find the volume of revolution shown on the right by rotating the area shown on the left about the x - axis…

Explanation:

Step1: Recall disk - method formula

The formula for the volume $V$ of the solid of revolution about the $x$-axis using the disk method is $V=\pi\int_{a}^{b}[f(x)]^{2}dx$, where $y = f(x)$ is the function and $[a,b]$ is the interval of integration.

Step2: Identify the function and interval

Here, $f(x)=\sqrt{\cos(x)}$, $a =-\frac{\pi}{2}$, and $b=\frac{\pi}{2}$.

Step3: Substitute into the formula

$V=\pi\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(\sqrt{\cos(x)})^{2}dx=\pi\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos(x)dx$.

Step4: Evaluate the integral

Since the antiderivative of $\cos(x)$ is $\sin(x)$, we have $\pi[\sin(x)]_{-\frac{\pi}{2}}^{\frac{\pi}{2}}$.

Step5: Calculate the definite - integral value

$\pi\left[\sin\left(\frac{\pi}{2}\right)-\sin\left(-\frac{\pi}{2}\right)\right]=\pi(1 - (- 1))=2\pi\approx6.283$.

Answer:

$6.283$