I am getting very confused about the following. Let's say I want to find the volume of a sphere. I can start with a circle having circumference $2\pi R\cos\theta$. I can multiply by $R d\theta$ and integrate to get $4\pi R^2$ as the surface area. Then, I can convert $R$ to $r$ integrate from $0$ to $R$. to get the volume. That makes sense to me.
Now, however, let's try to do the same thing with discs. I have a disk with area $\pi R^2 \sin^2\theta$. It seems like I should add a bunch of discs with height $R d\theta$, but that doesn't work. I really need to add discs with height $R\sin\theta d\theta$ (in order to get it to come out right), yet I do not have a good understanding of why. It isn't 100% obvious to me why $R \sin\theta d\theta$ is the correct height. I can sort of see where the pieces of it come from, but certainly could not draw a picture, for example.
with the discs the area is $\pi r^2 \cos^2 \theta$, and the thickness is $r \cos\theta d\theta$ so you want: $$ \frac{V}{\pi r^3} = \int_{-\frac{\pi}2}^{\frac{\pi}2} \cos^3 \theta d\theta = \int_{-\frac{\pi}2}^{\frac{\pi}2}(1-\sin^2 \theta)\cos \theta $$