Question

Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded by:

Answer 1

The volume is

$\displaystyle2\pi\int_0^2x^3\,\mathrm dx=\frac\pi2 x^4\bigg|_0^2=8\pi$

Each shell has a height of $x^2$ (distance between $y=x^2$ and $y=0$) and width equal to the distance to the axis of revolution ($x-0=x$), hence contributing a surface area of $2\pi x^3$. Sum (integrate) over all $x$ in the region of interest to get the volume.