Question

Find the volume of the solid whose base is the region in the first quadrant bounded by y=x3, y=1, and the y-axis and whose cross-sections perpendicular to the x axis are semicircles.

Answer 1

For a given $x$ in the interval [0, 1], the corresponding cross-section is a semicircle with diameter $d=1-x^3$, so the area of the cross-section would be $\frac{\pi d^2}8$.

The volume of the solid is then

$\displaystyle\frac\pi8\int_0^1(1-x^3)^2\,\mathrm dx=\boxed{\frac{9\pi}{112}}$