Chapter 6.4, Problem 47E

To determine

The area of the shaded region.

Answer to Problem 47E

For shaded region,

  $\text{Area}=\pi$

Explanation of Solution

Given information:

  

First

We have to find the area under the graph of $y=1+\cos x$ (Area of the non − shaded region).

  ${\displaystyle {\int }_0^{\pi }\left(1+\cos x\right)dx}={x+\sin x|}_0^{\pi }=\pi$

Then

Find the area of rectangle.

  $\text{Area}\text{of}\text{rectangle}=y\times x=2\times \pi =2\pi$

Now,

To find the area of the shaded region, we have to subtract the area of non − shaded region from the area of rectangle:

  $\begin{array}{l}\text{Area}\text{of}\text{the}\text{shaded}\text{region}=\text{Area}\text{of}\text{rectangle}-\text{Area}\text{of the non-shaded region}\\ =2\pi -\pi \\ =\pi \end{array}$