Top integral should be cos^3 not squared and you forget a r. Those two integrals aren't the same otherwise.
And the volume of a sphere is
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.
Actually, you may be right. Do you by any chance know the proper substitution? The first integral was essentially a "washers" integral using as the variable of integration. I'm actually not sure if is the washers or shells integral that results after the substitution.
I changed the 2nd integral in my post to the shells integral, btw.
Thanks so much for helping out!
Hmmm... I distinctly remember my 1st integral the one we used in class and it worked, but I think you are showing me what I should do for the shells method. I think I need the washers method actually.
EDIT: I just saw your reply above, thanks for helping me understand shells using so much better. But I was wondering if you could still show me washers- thats the one I initially wanted.