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evablogger [386]
2 years ago
13

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

Mathematics
1 answer:
BartSMP [9]2 years ago
8 0

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 <em>y</em> = 9 sin(<em>x</em>) and hence area = <em>π</em> (9 sin(<em>x</em>))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}

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