find the volume of the solid generated by revolving the following region about the given axis. the region in…

Explanation:

Step1: Recall disk - method formula

The volume $V$ of the solid of revolution about the $x$-axis using the disk method is given by $V=\pi\int_{a}^{b}[f(x)]^{2}dx$, where $y = f(x)$ is the curve being revolved and $[a,b]$ is the interval of $x$ - values. Here, $f(x)=x^{2}$, $a = 0$, and $b = 2$.

Step2: Set up the integral

$V=\pi\int_{0}^{2}(x^{2})^{2}dx=\pi\int_{0}^{2}x^{4}dx$.

Step3: Integrate $x^{4}$

Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $\int x^{4}dx=\frac{x^{5}}{5}+C$.

Step4: Evaluate the definite integral

$V=\pi\left[\frac{x^{5}}{5}\right]_{0}^{2}=\pi\left(\frac{2^{5}}{5}-\frac{0^{5}}{5}\right)$.

Step5: Calculate the result

$V=\pi\times\frac{32}{5}=\frac{32\pi}{5}$.

Answer:

$\frac{32\pi}{5}$