Solid of revolution vs $\iiint$

Question

In calc 1, I learned about rotating a curve around an axis, say $y=x^2$ around the y-axis. In calc 3, I learned about the shape of 3D objects in the context of $\iiint$ triple integrals . These concepts seem very related. No one, however, have connected them for me. Are they related, and if so: how?

Answer 1

In cylindrical co-ordinates (height along central axis $x$, distance from central axis $r$ and angle $\theta$), let the shape be bounded by $$0\le x \le x_0$$ $$0\le r \le r_0(x)$$ $$0\le\theta<2\pi$$ So the solid of revolution is given by $$V=\int_0^{x_0}\int_{0}^{r_0(x)}\int_0^{2\pi}dV=\int_0^{x_0}\int_{0}^{r_0(x)}\int_0^{2\pi}r d\theta dr dx$$ $$V=\int_0^{x_0}\int_{0}^{r_0(x)}2\pi r dr dx$$ $$V=\int_0^{x_0}\pi r_0(x)^2 dx,$$ which is the usual formula for volume of relvotution given the radius-function $r_0(x)$.