Symbolic integration has 3 solutions based on integration variable range, how to extract one of these?

r is the integration variable. The integration is:

mom_2 = int((umax*(1-r/Radius)^(1/7))^2*2*pi*r,r,0,Radius)

and the result is:

size(mom_2) = 1 1

Question: how do I access each of these three possible solutions?

For example, I would like to use (50/49)*pi*U_0^2 in further calculations. Thanks ahead of time!

Answer 1

[Paul]

syms U_0 r Radius
mom_2 = int((U_0*(1-r/Radius)^(1/7))^2*2*sym(pi)*r,r,0,Radius)

mom_2 = 

One approach that just extracts the case you want

c = children(mom_2)

c = 3×2 cell array
    {[(49*pi*U_0^2)/72                                    ]}    {[Radius == 1]}
    {[(49*pi*Radius^2*U_0^2)/72                           ]}    {[0 < Radius ]}
    {[2*U_0^2*pi*int(r*(1 - r/Radius)^(2/7), r, 0, Radius)]}    {[~0 < Radius]}

case1 = c{1,1}

case1 = 

[Jason Oakley]

Awesome response, and so quick too!

Answer 2

[John D'Errico]

Or do this:

syms r umax Radius
mom_2 = int((umax*(1-r/Radius)^(1/7))^2*2*pi*r,r,0,Radius)

mom_2 = 

subs(mom_2,Radius,1)

ans = 

Note that it resolves the three cases into 1.

[Paul]

It becomes more problematic to pull out the second case while keeping Radius in the result. The problem is with the third case ( which I guess is trying to account for some limiting situtation, using Symbolic notation I've not seen before) and coming up with a way to disambiguate it from the second case.

syms r umax Radius
mom_2 = int((umax*(1-r/Radius)^(1/7))^2*2*pi*r,r,0,Radius)

mom_2 = 

If we want just the second case, we could try

assume([Radius > 0, Radius ~= 1])
simplify(mom_2)

ans = 

assumptions('clear');

But how to specify anything about Radius that get rid of that third case?

Of coure, we can see by inspection that the second case can be obatined by

subs(mom_2,Radius,1)*Radius^2

ans = 

But that might be difficult to do in more complicated situations.

[Jason Oakley]

John, thank you since yours works too.

This is for fluid flow in a circular pipe. Physically, the radius is always R>0, so the 2nd and 3rd solutions are not applicable.

[Jason Oakley]

I think the first solution should have been for the general case R>1 instead of just R=1. Then the Radius^2 would be included in the result instead of adding it instead of needing to add it. No matter what the Radius is we could always nondimensionalize it so it is one and then the first case for R=1 is just fine.

[Paul]

It's usually better to use assume before calling int to give it some help. In this case, if R>0, we'd try

syms U_0 r Radius
assume(Radius,'positive')
mom_2 = int((U_0*(1-r/Radius)^(1/7))^2*2*sym(pi)*r,r,0,Radius)

mom_2 = 

Not sure why that didn't work. Instead, we can do

syms rho
assume(rho,'positive')
mom_2 = int((U_0*(1-r/rho)^(1/7))^2*2*sym(pi)*r,r,0,Radius)

mom_2 = 

mom_2 = subs(mom_2,rho,Radius)

mom_2 = 

[Jason Oakley]

Yes, that is a good tip and now this looks like the cleanest solution