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,
does not work.
I believe my teacher explained that it had to do with the that were in fact small wedges that, in sum, did not produce a sphere. However, the substitution using and did produce the correct result, which is actually the same as the conventionally derived (single-variable) shells integral for a sphere:
Now, shouldn't these integrals be the same, despite the substitution? What was the first integral really solving 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.