Chapter 7, Problem 53EP

a.

To determine

To find: The area of the region bounded by given equation.

Answer to Problem 53EP

The area $A=1.36604$ .

Explanation of Solution

Given information:

  $\begin{array}{l}y=2+\sin x\\ y=\sec x\end{array}$

Calculation:

Let's graph the area to draw some conclusions about it.

  

To find the anti-derivative for $\sec x$ ,

  $\begin{array}{c}\sec \left(x\right)=\frac{\sec \left(x\right)\left(\sec \left(x\right)+\tan \left(x\right)\right)}{\sec \left(x\right)+\tan \left(x\right)}\\ \frac{d\left(\sec \left(x\right)+\tan \left(x\right)\right)}{dx}=\sec \left(x\right)\left(\sec \left(x\right)+\tan \left(x\right)\right)\end{array}$

The general formula for the above equation is,

  ${\displaystyle \int \frac{dt}{t}=\ln \left|t\right|}+C$

Now, the integral becomes,

  ${\displaystyle \int \sec \left(x\right)dx=\ln \sec \left(x\right)+\tan \left(x\right)}+C$

Finding the area by using the Fundamental Theorem of calculus,

  $\begin{array}{c}A={\displaystyle \underset{0}{\overset{1.224}{\int }}\left(2+\sin \left(x\right)-\sec \left(x\right)\right)dx}\\ A={\left.\left(2x-\cos \left(x\right)-\ln \left|\sec \left(x\right)+\tan \left(x\right)\right|\right)\right|}_0^{1.224}\\ A=1.36604\end{array}$

Therefore, the area $A=1.36604$ .

b.

To determine

To find: The volume of the solid generated when $R$ is revolved about the $x$ -axis.

Answer to Problem 53EP

The volume of the solid generated is $V=16.4042$ .

Explanation of Solution

Given information:

  $\begin{array}{l}y=2+\sin x\\ y=\sec x\end{array}$

Calculation:

In order to compute the volume, first determine the area $A\left(x\right)$ of the cross-section of the solid at height $x\in \left[0,1.224\right]$ .

  $V={\displaystyle \underset{0}{\overset{1.224}{\int }}A\left(x\right)}dx$

A circular portion of radius $2+\sin \left(x\right)$ and a circular region of radius $\sec \left(x\right)$ are cut from the center of the solid perpendicular to the $x$ -axis on height $x\in \left[0,1.224\right]$ . The area $A\left(x\right)$ is therefore given as,

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

Finding the volume by using the Fundamental Theorem of calculus,

  $\begin{array}{c}V={\displaystyle \underset{0}{\overset{1.224}{\int }}\left(4+4\sin \left(x\right)+{\sin }^2\left(x\right)-{\sec }^2\left(x\right)\right)\pi dx}\\ ={\left.\left(4x-4\cos \left(x\right)+\frac{x}{2}-\frac{\sin \left(2x\right)}{4}-\tan \left(x\right)\right)\right|}_0^{1.224}\\ V=16.4042\end{array}$

Therefore, the volume of the solid generated is $V=16.4042$ .

c.

To determine

To find: The volume of the solid whose base is $R$ and whose cross sections cut by planes perpendicular to the $x$ -axis are squares.

Answer to Problem 53EP

The volume of the solid is $V=1.62906$ .

Explanation of Solution

Given information:

  $\begin{array}{l}y=2+\sin x\\ y=\sec x\end{array}$

Calculation:

In order to compute the volume, first determine the area $A\left(x\right)$ of the cross-section of the solid at height $x\in \left[0,1.224\right]$ .

  $V={\displaystyle \underset{0}{\overset{1.224}{\int }}A\left(x\right)}dx$

To solve the integral in order to determine the antiderivative for the function $\tan \left(x\right)$ .

Finding the volume by using the Fundamental Theorem of calculus,

  $\begin{array}{c}V={\displaystyle \underset{0}{\overset{1.224}{\int }}\left(4+{\sin }^2\left(x\right)+{\sec }^2\left(x\right)+4\sin \left(x\right)-4\sec \left(x\right)-2\tan \left(x\right)\right)dx}\\ V={\left.\left(4x+\frac{x}{2}-\frac{\sin \left(2x\right)}{4}+\tan \left(x\right)-4\cos \left(x\right)-4\ln \left|\sec \left(x\right)\right|+2\ln \left|\sin \left(x\right)\right|\right)\right|}_0^{1.224}\\ V=1.62906\end{array}$

Therefore, the volume of the solid generated is $V=1.62906$ .