### Problem 16Find the volume obtained by rotating the curve $ y = \sin^2 x $ around the $ x $-axis.

Answered: Find the volume obtained by rotating…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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### Problem 16

Find the volume obtained by rotating the curve $ y = \sin^2 x $ around the $ x $-axis.

Expert Solution

Step 1

The volume of the solid formed by rotating the curve $y = \sin^{2} x$ around the x -axis is given by $\int_{a}^{b} π {(f (x))}^{2} dx$

Substitute the values

$\begin{array}{rcl} \int_{a}^{b} π {(f (x))}^{2} dx & = & \int_{0}^{π} π {(\sin^{2} x)}^{2} dx \\ = & \int_{0}^{π} π (\sin^{4} x) dx \end{array}$

$\begin{array}{rcl} \int_{a}^{b} π {(f (x))}^{2} dx & = & π \int_{0}^{π} (\sin^{4} x) dx \end{array}$ ...... (1)

using the formula $\int \sin^{n} x d x = - \frac{\cos x \sin^{n - 1} x}{n} + \frac{n - 1}{n} \int \sin^{n - 2} x d x$

$\begin{array}{rcl} \int \sin^{4} x d x & = & - {[\frac{\cos x \sin^{4 - 1} x}{4}]}_{0}^{π} + \frac{4 - 1}{4} \int_{0}^{π} \sin^{4 - 2} x d x \\ = & - {[\frac{\cos x \sin^{3} x}{4}]}_{0}^{π} + \frac{3}{4} \int_{0}^{π} \sin^{2} x d x \\ = & - {[\frac{\cos x \sin^{3} x}{4}]}_{0}^{π} + \frac{3}{4} \int_{0}^{π} \frac{(1 - \cos 2 x)}{2} d x \\ = & - {[\frac{\cos x \sin^{3} x}{4}]}_{0}^{π} + \frac{3}{8} {[x - \frac{\sin 2 x}{2}]}_{0}^{π} \\ = & - [\frac{cosπ \sin^{3} π}{4} - \frac{\cos 0 \sin^{3} 0}{4}] + \frac{3}{8} [π - \frac{\sin 2 π}{2} - 0] \\ = & 0 + \frac{3}{8} [π] \\ = & \frac{3 π}{8} \end{array}$

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ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated