Question

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

Answer 1

Using the disk method, the volume is given by the integral

`\displaystyle \pi \int_(\pi/2)^\pi (9\sin(x))^2\,\mathrm dx = 81\pi \int_(\pi/2)^\pi \sin^2(x)\,\mathrm dx`

That is, each disk has a radius of y = 9 sin(x) and hence area = π (9 sin(x))². Add up infinitely many such disks by integrating. Then the volume is

`\displaystyle 81\pi \int_(\pi/2)^\pi \sin^2(x)\,\mathrm dx = \frac{81\pi}2 \int_(\pi/2)^\pi (1-\cos(2x))\,\mathrm dx \\\\ =\frac{81\pi}2 \left(x-\frac{\sin(2x)}2\right)\bigg|_(\pi/2)^\pi \\\\ = \frac{81\pi}2 \left( \left(\pi-\frac{\sin(2\pi)}2\right) - \left(\frac\pi2 - \frac{\sin(\pi)}2\right) \right) \\\\ = \boxed{\frac{81\pi^2}4}`