Let [math]R[/math] be the region bounded by [math]y=x^2, y=4[/math] and the [math]y[/math]-axis. If the volume of the solid of revolution around [math]y[/math]-axis is [math]a\pi[/math] where [math]a[/math] is a rational number, what is [math]a[/math]?

Let [math]R[/math] be the region bounded by [math]y=4-x, y=2[/math] and the [math]y[/math]-axis. If the volume of the solid of revolution around [math]x=4[/math] is [math]a\pi[/math] where [math]a[/math] is a rational number, what is [math]a[/math]?

Let [math]R=\{(x, y)|1\leq(x-2)^2+y^2\leq4\}[/math]. Let [math]V[/math] be the solid formed by the region [math]R[/math] rotating [math]180^\circ[/math] around the [math]y[/math]-axis. Write, but do not solve, the volume of the solid.

(Calculator) Consider two function [math]f(x)=4(x-2)[/math] and [math]g(x)=-(x-3)^2+4[/math]. Let [math]R[/math] be the region bounded by [math]f(x)[/math] and [math]g(x)[/math].
(a) Find the area of [math]R[/math].
(b) Let [math]V[/math] the solid formed by rotating the region [math]R[/math] about the horizontal line [math]y=5[/math]. Write, but do not solve, the volume of [math]V[/math].
(c) Let [math]W[/math] the solid formed by rotating the region [math]R[/math] about [math]x=-2[/math]. Write, but do not solve, the volume of [math]W[/math].