Chapter 7.3, Problem 24E

To determine

To find: The volume of the solid.

Answer to Problem 24E

The volume of the given solid $V=\pi (3\pi -8)$ .

Explanation of Solution

Given information:

The line $y=2$

And the curve $y=2\sin x,0\le x\le \pi /2$ .

Calculation:

Must locate the equation's smallest solution in order to determine the limits of the integration.

  $2=2\sin (x)$

And the equation's smallest nonnegative solution:

  $0=2\sin (x)$

The limits of integration are represented by the smallest nonnegative solutions of this equation, which are $x=0$ and

  $x=\frac{\pi }{2}$

  $($ this is the procedure how to obtain the limits of integration which is given $)$ .

Each cross-section for $x\in \left[0,\frac{\pi }{2}\right]$ is a circular region of radius $2-2\sin (x)$ and the area of this cross-section is:

  $\begin{array}{c}$ A(x)={(2-2\sin (x))}^2\pi $=\left(4-8\sin (x)+4{\sin }^2(x)\right)\pi \\ =(4-8\sin (x)+2(1-\cos (2x)))\pi \\ =(6-8\sin (x)-2\cos (2x))\pi \end{array}$

Then used the identity

  ${\sin }^2(x)=\frac{1-\cos (2x)}{2}$

Consequently, the given solid's volume is given by

  

The given region revolving around the line $y=2$ to obtain the solid is shaded purple in the picture below.

  

Therefore, the required volume of the given solid $V=\pi (3\pi -8)$ .