Question

Find the volume V obtained by rotating the region bounded by the given curves about the specified axis.

y = 3 sin2x, y = 0, 0 ≤ x ≤ π; about the x−axis

Answer 1

The following is a function of x, and is to be rotated about the x-axis. The disk method is easiest to implement.

Answer 2

We may split the volume into infinitesimal disks with volume \(\mathrm{d}V\). To determine the volume of the disk, we use \(\pi r^2h\), where \(r=3\sin2x\) and \(h=\mathrm{d}x\). Thus to determine our entire volume, \(V\), we integrate the volume of our infinitesimal disks:
$$\begin{align*}V&=\int_0^\pi\mathrm{d}V\\&=\int_0^\pi\pi(3\sin2x)^2\,\mathrm{d}x\\&=9\pi\int_0^\pi\sin^22x\,\mathrm{d}x\\&=\frac92\pi\int_0^\pi(1-\cos4x)\,\mathrm{d}x\\&=\frac92\pi\left[x-\frac14\sin4x\right]_0^\pi\\&=\frac92\pi\left[\pi-\frac14\sin4\pi\right]\\&=\frac92\pi^2\\&\end{align*}$$