Solids of Revolution (idea)

OK, here's the shell method from somebody who aced AP calculus.

Shell method of integrating volumes of solids of revolution

The shell method is useful if y(r) is easier to integrate than r(y) (for instance, if the volume is rotated about the y-axis). Imagine a cylinder made of concentric shells (a scroll comes to mind). The volume of such a shell is

V = π(R² - r²)h

Approximating the volume by concentric shells:

V = ∑(π((r + Δr)² - r²)h(r))
= ∑(π(r² + 2rΔr + Δr² - r²)h(r))
= ∑(π(2rΔr + Δr^2)h(r))

Taking the limit as Δr goes to 0:

= ∫2πr h(r) ∂r

which seems intuitive, as 2πr h(r) is the area of such a shell.