Hello all,

So I'm trying to figure out why setting up integrals to find volume of a sphere with the correct variable of integration.

For example, for volume,

$\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \pi r^2 \cos^2(\theta)d\theta=\frac{\pi r^2}{4} $ does not work.

I believe my teacher explained that it had to do with the $\displaystyle d\theta $ that were in fact small wedges that, in sum, did not produce a sphere. However, the substitution using $\displaystyle x=r*\sin(\theta) $ and $\displaystyle dx=r*\cos(\theta)d\theta $ did produce the correct result, which is actually the same as the conventionally derived (single-variable) shells integral for a sphere:

$\displaystyle \pi \int_{-r}^{r} x \sqrt{r^2-x^2} dx $

Now, shouldn't these integrals be the same, despite the substitution? What was the first integralreallysolving for- I don't think I can picture it in my head.

Thank you for taking the time to read this post and help me out.