find the volume of the solid generated by revolving the region bounded by the given lines and curves about…

Explanation:

Step1: Recall the 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. Here, $f(x)=x$, $a = 2$, and $b = 4$.

Step2: Set up the integral

$V=\pi\int_{2}^{4}x^{2}dx$.

Step3: Integrate $x^{2}$

The antiderivative of $x^{2}$ is $\frac{1}{3}x^{3}$. So, $\pi\int_{2}^{4}x^{2}dx=\pi\left[\frac{1}{3}x^{3}\right]_{2}^{4}$.

Step4: Evaluate the definite - integral

$\pi\left(\frac{1}{3}(4)^{3}-\frac{1}{3}(2)^{3}\right)=\pi\left(\frac{64}{3}-\frac{8}{3}\right)=\pi\times\frac{64 - 8}{3}=\frac{56}{3}\pi$.

Answer:

A. $\frac{56}{3}\pi$