Compute Multiple Integrals

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To compute multiple integrals, use nested calls to int. For example, compute the surface area and the volume of a sphere using spherical coordinates.

Compute the expression for the surface area of a sphere. The distance from the center of a sphere to the surface remains constant, r = R. The angle ϕ changes its value from 0 to π. The angle χ changes its value from 0 to 2 π:

int(int(R^2*sin(phi), phi = 0..PI), chi = 0..2*PI)

Compute the expression for the volume of a sphere. The angles accept the same values, but the distance from the center of any point inside the sphere ranges from 0 to R. To find the volume, compute the following triple integral:

int(int(int(r^2*sin(phi), r = 0..R), phi = 0..PI), chi = 0..2*PI)