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Hi everyone. I am trying to integrate the following:

[itex]\int^{\frac{π}{2}}_{-\frac{π}{2}}\int^{acosθ}_{0}\int^{\sqrt{a^{2}-r^{2}}}_{-\sqrt{a^{2}-r^{2}}}rdzdrdθ[/itex]

Here's my work:

[itex]=2\int^{\frac{π}{2}}_{-\frac{π}{2}}\int^{acosθ}_{0}r\sqrt{a^{2}-r^{2}}drdθ[/itex]

I use substitution with u=a

[itex]=-\int^{\frac{π}{2}}_{-\frac{π}{2}}\int^{a^{2}sin^{2}θ}_{a^{2}}u^{\frac{1}{2}}dudθ[/itex]

[itex]=-\frac{2}{3}a^{3}\int^{\frac{π}{2}}_{-\frac{π}{2}}(sin^{3}θ-1)dθ[/itex]

[itex]=-\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[/itex]

sin

[itex]=\frac{2}{3}a^{3}\int^{\frac{π}{2}}_{-\frac{π}{2}}dθ[/itex]

[itex]=\frac{2}{3}πa^{3}[/itex]

However, I have seen other solutions online that give the actual answer as 2a

[itex]\int^{\frac{π}{2}}_{-\frac{π}{2}}\int^{acosθ}_{0}\int^{\sqrt{a^{2}-r^{2}}}_{-\sqrt{a^{2}-r^{2}}}rdzdrdθ[/itex]

Here's my work:

[itex]=2\int^{\frac{π}{2}}_{-\frac{π}{2}}\int^{acosθ}_{0}r\sqrt{a^{2}-r^{2}}drdθ[/itex]

I use substitution with u=a

^{2}-r^{2}to get:[itex]=-\int^{\frac{π}{2}}_{-\frac{π}{2}}\int^{a^{2}sin^{2}θ}_{a^{2}}u^{\frac{1}{2}}dudθ[/itex]

[itex]=-\frac{2}{3}a^{3}\int^{\frac{π}{2}}_{-\frac{π}{2}}(sin^{3}θ-1)dθ[/itex]

[itex]=-\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[/itex]

sin

^{3}θ is an odd function so the first integral is equal to zero:[itex]=\frac{2}{3}a^{3}\int^{\frac{π}{2}}_{-\frac{π}{2}}dθ[/itex]

[itex]=\frac{2}{3}πa^{3}[/itex]

However, I have seen other solutions online that give the actual answer as 2a

^{3}(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 sin^{3}θ, 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.
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