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## Homework Statement

The problem was to find the volume enclosed by a sphere of radius "a" centered on the origin by crafting a triple integral and solving for it using

**cylindrical**coordinates.

## Homework Equations

[tex]x^{2}+y^{2}+z^{2}=a^{2}[/tex] : Equation for a sphere of radius "a" centered on the origin.

[tex]\iiint\limits_E dV[/tex] : Triple integral for finding volume of a region [tex]E[/tex].

## The Attempt at a Solution

I solved the triple integral (but I don't think it's right) and got this: [tex]\frac{4}{3}a^{2}\pi[/tex] ---> Actually, I think I solved the integral right, but I think my bounds are incorrect.

I used the following as my bounds and subsequent iterated integral:

[tex]E=\{ \ (r,\theta,z) \ | \ 0\leq r\leq a, \ 0\leq \theta\leq 2\pi, \ -\sqrt{a^2-r^2}\leq z\leq \sqrt{a^2-r^2} \ \}[/tex]

[tex]\int^{2\pi}_{0}\int^{a}_{0}\int^{\sqrt{a^2-r^2}}_{\sqrt{a^2-r^2}} dz dr d\theta[/tex]

If my proposed answer isn't right could the problem lie within my bounds? I'm not really great at determining the bounds for iterated integrals yet >.<'

Thanks :D