# Triple Integral

Hi everyone. I am trying to integrate the following:

$\int^{\frac{π}{2}}_{-\frac{π}{2}}\int^{acosθ}_{0}\int^{\sqrt{a^{2}-r^{2}}}_{-\sqrt{a^{2}-r^{2}}}rdzdrdθ$

Here's my work:

$=2\int^{\frac{π}{2}}_{-\frac{π}{2}}\int^{acosθ}_{0}r\sqrt{a^{2}-r^{2}}drdθ$

I use substitution with u=a2-r2 to get:

$=-\int^{\frac{π}{2}}_{-\frac{π}{2}}\int^{a^{2}sin^{2}θ}_{a^{2}}u^{\frac{1}{2}}dudθ$

$=-\frac{2}{3}a^{3}\int^{\frac{π}{2}}_{-\frac{π}{2}}(sin^{3}θ-1)dθ$

$=-\frac{2}{3}a^{3}\int^{\frac{π}{2}}_{-\frac{π}{2}}sin^{3}θdθ+\frac{2}{3}a^{3}\int^{\frac{π}{2}}_{-\frac{π}{2}}dθ$

sin3θ is an odd function so the first integral is equal to zero:

$=\frac{2}{3}a^{3}\int^{\frac{π}{2}}_{-\frac{π}{2}}dθ$

$=\frac{2}{3}πa^{3}$

However, I have seen other solutions online that give the actual answer as 2a3(3π-4)/9. The authors of these solutions change the limits of integration of the original triple integral by taking advantage of symmetry. More specifically, I noticed that the person changed the limits of θ from 0 to π/2 by multiplying the integral by 2. Then when you get to the point when you integrate sin3θ, the integral no longer equals zero. However, I thought you could only do this if the region of integration has no θ dependence. And in any case, why should it matter whether I change the limits of integration or not? It's still the same integral, right? I have a feeling I'm making a very dumb mistake somewhere but I can't find out where. Thanks for any help.

Last edited:

Related Calculus and Beyond Homework Help News on Phys.org
DryRun
Gold Member
sin3θ is an odd function so the first integral is equal to zero:
You could also have used the multiple angle formula to convert sin3θ into sin(3θ) or used the substitution: let u = cosθ.

LCKurtz
Homework Helper
Gold Member
Hi everyone. I am trying to integrate the following:

$\int^{\frac{π}{2}}_{-\frac{π}{2}}\int^{acosθ}_{0}\int^{\sqrt{a^{2}-r^{2}}}_{-\sqrt{a^{2}-r^{2}}}rdzdrdθ$

Here's my work:

$=2\int^{\frac{π}{2}}_{-\frac{π}{2}}\int^{acosθ}_{0}r\sqrt{a^{2}-r^{2}}drdθ$

I use substitution with u=a2-r2 to get:

$=-\int^{\frac{π}{2}}_{-\frac{π}{2}}\int^{a^{2}sin^{2}θ}_{a^{2}}u^{\frac{1}{2}}dudθ$

$=-\frac{2}{3}a^{3}\int^{\frac{π}{2}}_{-\frac{π}{2}}(sin^{3}θ-1)dθ$

$=-\frac{2}{3}a^{3}\int^{\frac{π}{2}}_{-\frac{π}{2}}sin^{3}θdθ+\frac{2}{3}a^{3}\int^{\frac{π}{2}}_{-\frac{π}{2}}dθ$

sin3θ is an odd function so the first integral is equal to zero:

$=\frac{2}{3}a^{3}\int^{\frac{π}{2}}_{-\frac{π}{2}}dθ$

$=\frac{2}{3}πa^{3}$

However, I have seen other solutions online that give the actual answer as 2a3(3π-4)/9. The authors of these solutions change the limits of integration of the original triple integral by taking advantage of symmetry. More specifically, I noticed that the person changed the limits of θ from 0 to π/2 by multiplying the integral by 2. Then when you get to the point when you integrate sin3θ, the integral no longer equals zero. However, I thought you could only do this if the region of integration has no θ dependence. And in any case, why should it matter whether I change the limits of integration or not? It's still the same integral, right? I have a feeling I'm making a very dumb mistake somewhere but I can't find out where. Thanks for any help.
You didn't make a dumb mistake, but it is a mistake nevertheless, sort of a subtle one. The problem is where you have $$u^{\frac 3 2}|_{a^2}^{a^2\sin^2\theta}$$and you substitute the ##a^2\sin^2\theta## in for the ##u##. That gives you $$(a^2\sin^2\theta)^{\frac 3 2}$$That does not simplify to ##a^3\sin^3\theta## when ##\theta## is between ##-\frac \pi 2## and ##0## because ##a^2\sin^2\theta\ge 0## and when you raise it to the 3/2 power it must be nonnegative. It's the old problem that ##\sqrt{x^2}= |x|##, not ##x##. What it does simplify to is ##a^3|\sin^3(\theta)|##, which is not an odd function and doesn't give ##0## in later steps.

Thanks LCKurtz. That ,makes sense. Now what I don't understand is this:

$\int^{\frac{π}{2}}_{-\frac{π}{2}}\int^{acosθ}_{0}\int^{\sqrt{a^{2}-r^{2}}}_{-\sqrt{a^{2}-r^{2}}}rdzdrdθ=4\int^{\frac{π}{2}}_{0}\int^{acosθ}_{0}\int^{\sqrt{a^{2}-r^{2}}}_{0}rdzdrdθ$

It makes sense to me with the integral with respect to z, since the region of integration is symmetrical about the xy-plane, and f(θ,r,z)=r is even with respect to z. As for the integral with respect to θ, I realize that the region of integration is also symmetrical about the yz-plane, but I am not immediately inclined to think that the function f(θ) being integrated at that time will be even with respect to θ. How can you tell? Am I even thinking of symmetry in the right way?

EDIT: Is it because g(θ)=acosθ is even with respect to θ, so any composite function f(g(θ)) that results from the integration with respect to r will also be even?

Last edited:
LCKurtz
Homework Helper
Gold Member
Thanks LCKurtz. That ,makes sense. Now what I don't understand is this:

$\int^{\frac{π}{2}}_{-\frac{π}{2}}\int^{acosθ}_{0}\int^{\sqrt{a^{2}-r^{2}}}_{-\sqrt{a^{2}-r^{2}}}rdzdrdθ=4\int^{\frac{π}{2}}_{0}\int^{acosθ}_{0}\int^{\sqrt{a^{2}-r^{2}}}_{0}rdzdrdθ$

It makes sense to me with the integral with respect to z, since the region of integration is symmetrical about the xy-plane, and f(θ,r,z)=r is even with respect to z. As for the integral with respect to θ, I realize that the region of integration is also symmetrical about the yz-plane, but I am not immediately inclined to think that the function f(θ) being integrated at that time will be even with respect to θ. How can you tell? Am I even thinking of symmetry in the right way?

EDIT: Is it because g(θ)=acosθ is even with respect to θ, so any composite function f(g(θ)) that results from the integration with respect to r will also be even?
##\cos\theta## is even and any function of an even function is even. See if you can prove that.

Let x=g(t) be an even function, that is, x=g(t)=g(-t). Then f(x)=f(g(t))=f(g(-t)) must also be an even function.

Is that a sufficient proof? It sounds more like a conclusion.

LCKurtz
Homework Helper
Gold Member
Let x=g(t) be an even function, that is, x=g(t)=g(-t). Then f(x)=f(g(t))=f(g(-t)) must also be an even function.

Is that a sufficient proof? It sounds more like a conclusion.
That's about all there is to it. I would write it like this: Let ##h(t) = f(g(t))##. You want to show ##h(t) = h(-t)##. So calculate ##h(-t)##:
##h(-t) = f(g(-t)) = f(g(t))\hbox{ (since g is even) } = h(t)##.

Okay, thanks again. This problem was driving me insane before but it's all clear now.