The base of the solid is the region between the $x$-axis and the parabola $y=4-x^2$. The vertical cross sections of the solid perpendicular to the $y$-axis are semicircles. Compute the volume of the solid.
I know the area formula for a semicircle is $\frac{(\pi*r^2)}{2}$, but would you use $r=4 – x^2$ even though the question only says the base is made from this parabola?
Best Answer
Each semicircle has a radius $x$, so the solid has volume
$$\frac{\pi}{2} \int_0^2 dy \, x^2 = \frac{\pi}{2} \int_0^2 dy \, (4-y) = 3 \pi$$