Problem 58259. Easy Sequences 115: Integral involving square root, floor, and round functions

Created by Ramon Villamangca

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Given a postive real number </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, we are asked to evaluate the following integral:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                            </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>s=S(n)=\\Bigg\\lfloor\\ \\int_0^n\\sqrt{\\lfloor{x}\\rfloor} - \\Big\\lfloor{\\sqrt{x} \\Big\\rfloor\\ \\mathrm{d}x\\  \\Bigg\\rceil \\cdot</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                            where:     the symbol \"</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\lfloor\\ \\rfloor</w:t></w:r></w:customXml><w:r><w:t>\" is the </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/help/matlab/ref/floor.html#\"><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>floor</w:t></w:r></w:hyperlink><w:r><w:t> function,</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                                            and \"</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\lfloor\\ \\rceil</w:t></w:r></w:customXml><w:r><w:t>\" is the </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/help/matlab/ref/round.html\"><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>round </w:t></w:r></w:hyperlink><w:r><w:t>(to the nearest integer) function.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>We may rewrite the above function in Matlab as:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[>>  S = @(n) round(integral(@(x) sqrt(floor(x))-floor(sqrt(x)),0,n));]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Therefore, for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n=10\\pi</w:t></w:r></w:customXml><w:r><w:t>, we have:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[>>  s = S(10*pi)\ns =\n        12]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Be careful though, in using the Matlab </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/help/matlab/ref/integral.html\"><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>integral </w:t></w:r></w:hyperlink><w:r><w:t>function, as it is only an approximation. For example if </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n=100000</w:t></w:r></w:customXml><w:r><w:t>:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[>>  s = S(100000)\n\nWarning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is\n8.0e+01. The integral may not exist, or it may be difficult to approximate numerically to the\nrequested accuracy. \n> In integralCalc/iterateScalarValued (line 372)\nIn integralCalc/vadapt (line 132)\nIn integralCalc (line 75)\nIn integral (line 87)\nIn @(n)round(integral(@(x)sqrt(floor(x))-floor(sqrt(x)),0,n)) \n\ns =\n       49841]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The correct answer is </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>s=49839</w:t></w:r></w:customXml><w:r><w:t>. The </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>integral</w:t></w:r><w:r><w:t> function is off by only 2 units here, but the discrepancy could be even greater for other values of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t>. </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>integral</w:t></w:r><w:r><w:t> function can also be quite slow. The challenge is to find an efficient and more accurate algorithm to evaluate the integral.</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; "><div style="block-size: 673px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 445.71875px 336.5px; transform-origin: 445.71875px 336.5px; vertical-align: baseline; "><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">Given a postive real number </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">, we are asked to evaluate the following integral:</span></span></div><div style="block-size: 48px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 24px; text-align: left; transform-origin: 384px 24px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                            </span></span><span style="vertical-align:-18px"><img 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" width="247.5" height="48" style="width: 247.5px; height: 48px;"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                            where:     the symbol "</span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="23.5" height="19" style="width: 23.5px; height: 19px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">" is the </span></span><a target='_blank' href = "https://www.mathworks.com/help/matlab/ref/floor.html#"><span style=""><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-weight: 700; ">floor</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> function,</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                                            and "</span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="23.5" height="19" style="width: 23.5px; height: 19px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">" is the </span></span><a target='_blank' href = "https://www.mathworks.com/help/matlab/ref/round.html"><span style=""><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-weight: 700; ">round </span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">(to the nearest integer) function.</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">We may rewrite the above function in Matlab as:</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 20px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 442.71875px 10px; transform-origin: 442.71875px 10px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">&gt;&gt;  S = @(n) round(integral(@(x) sqrt(floor(x))-floor(sqrt(x)),0,n));</span></span></div></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Therefore, for </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="53" height="18" style="width: 53px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, we have:</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 60px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 442.71875px 30px; transform-origin: 442.71875px 30px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 442.71875px 10px; transform-origin: 442.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">&gt;&gt;  s = S(10*pi)</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 442.71875px 10px; transform-origin: 442.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">s =</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 442.71875px 10px; transform-origin: 442.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">        12</span></span></div></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Be careful though, in using the Matlab </span></span><a target='_blank' href = "https://www.mathworks.com/help/matlab/ref/integral.html"><span style=""><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-weight: 700; ">integral </span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">function, as it is only an approximation. For example if </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="74" height="18" style="width: 74px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">:</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 260px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 442.71875px 130px; transform-origin: 442.71875px 130px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 442.71875px 10px; transform-origin: 442.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">&gt;&gt;  s = S(100000)</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 442.71875px 10px; transform-origin: 442.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; "></span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 442.71875px 10px; transform-origin: 442.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">Warning: Reached the </span><span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); ">limit on the maximum number of intervals in use. Approximate bound on error is</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 442.71875px 10px; transform-origin: 442.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">8.0e+01. The integral </span><span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); ">may not exist</span><span style="margin-inline-end: 0px; margin-right: 0px; ">, or </span><span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); ">it may be difficult to approximate numerically to the</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 442.71875px 10px; transform-origin: 442.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">requested </span><span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); ">accuracy. </span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 442.71875px 10px; transform-origin: 442.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">&gt; In integralCalc/iterateScalarValued (line 372)</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 442.71875px 10px; transform-origin: 442.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">In </span><span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); ">integralCalc/vadapt (line 132)</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 442.71875px 10px; transform-origin: 442.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">In </span><span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); ">integralCalc (line 75)</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 442.71875px 10px; transform-origin: 442.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">In </span><span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); ">integral (line 87)</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 442.71875px 10px; transform-origin: 442.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">In </span><span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); ">@(n)round(integral(@(x)sqrt(floor(x))-floor(sqrt(x)),0,n)) </span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 442.71875px 10px; transform-origin: 442.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; "></span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 442.71875px 10px; transform-origin: 442.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">s =</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 442.71875px 10px; transform-origin: 442.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">       49841</span></span></div></div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">The correct answer is </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="65" height="18" style="width: 65px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">. The </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-weight: 700; ">integral</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> function is off by only 2 units here, but the discrepancy could be even greater for other values of </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">. </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-weight: 700; ">integral</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> function can also be quite slow. The challenge is to find an efficient and more accurate algorithm to evaluate the integral.</span></span></div></div></div>

Given a postive real number , we are asked to evaluate the following integral:
                            
                            where:     the symbol "" is the floor function,
                                            and "" is the round (to the nearest integer) function.
We may rewrite the above function in Matlab as:
>>  S = @(n) round(integral(@(x) sqrt(floor(x))-floor(sqrt(x)),0,n));
Therefore, for , we have:
>>  s = S(10*pi)
s =
        12
Be careful though, in using the Matlab integral function, as it is only an approximation. For example if :
>>  s = S(100000)

Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is
8.0e+01. The integral may not exist, or it may be difficult to approximate numerically to the
requested accuracy. 
> In integralCalc/iterateScalarValued (line 372)
In integralCalc/vadapt (line 132)
In integralCalc (line 75)
In integral (line 87)
In @(n)round(integral(@(x)sqrt(floor(x))-floor(sqrt(x)),0,n))

s =
       49841
The correct answer is . The integral function is off by only 2 units here, but the discrepancy could be even greater for other values of . integral function can also be quite slow. The challenge is to find an efficient and more accurate algorithm to evaluate the integral.

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1 Comment

Ramon Villamangca on 18 May 2023

Hi David,

You right I should have use floor (34) instead of round (35). Test suites were adjusted. Thanks.

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Problem 58259. Easy Sequences 115: Integral involving square root, floor, and round functions

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