finding the volume of a solid of revolution.\n1. sketch the area bounded by the equations. y = x², y = 0, x…

Explanation:

Step1: Identify the radius of the cross - section

The region is bounded by $y = x^{2}$, $y = 0$, $x = 2$ and revolved around the $x$-axis. The radius of the circular cross - section at a given $x$ is $r=y=x^{2}$.

Step2: Recall the area formula for a circle

The area formula for a circle is $A=\pi r^{2}$.

Step3: Find the area of one cross - section

Substitute $r = x^{2}$ into the area formula. So, $A(x)=\pi(x^{2})^{2}=\pi x^{4}$.

Step4: Use the disk method to find the volume

The disk method formula for the volume of a solid of revolution about the $x$-axis is $V=\int_{a}^{b}A(x)dx$. Here, $a = 0$, $b = 2$ and $A(x)=\pi x^{4}$. So, $V=\int_{0}^{2}\pi x^{4}dx$.

Step5: Integrate the function

We know that $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). Then $\int_{0}^{2}\pi x^{4}dx=\pi\left[\frac{x^{5}}{5}\right]_{0}^{2}$.

Step6: Evaluate the definite integral

$\pi\left[\frac{x^{5}}{5}\right]_{0}^{2}=\pi\left(\frac{2^{5}}{5}-\frac{0^{5}}{5}\right)=\frac{32\pi}{5}$.

Answer:

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