use the shell method to find the volume of the solid generated by revolving the shaded region about the x…

Explanation:

Step1: Recall shell - method formula

The shell - method formula for volume when rotating about the x - axis is $V = 2\pi\int_{c}^{d}y\cdot h(y)dy$, where $y$ is the distance from the axis of rotation, and $h(y)$ is the height of the shell at $y$. Here, $c = 0$, $d=\sqrt{2}$, and $h(y)=(2 - y^{2})$.

Step2: Set up the integral

$V = 2\pi\int_{0}^{\sqrt{2}}y(2 - y^{2})dy=2\pi\int_{0}^{\sqrt{2}}(2y - y^{3})dy$.

Step3: Integrate term - by - term

Using the power rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $\int(2y - y^{3})dy=y^{2}-\frac{y^{4}}{4}+C$.

Step4: Evaluate the definite integral

$V = 2\pi\left[y^{2}-\frac{y^{4}}{4}\right]_{0}^{\sqrt{2}}$. Substitute the upper and lower limits: $V = 2\pi\left[\left((\sqrt{2})^{2}-\frac{(\sqrt{2})^{4}}{4}\right)-\left(0^{2}-\frac{0^{4}}{4}\right)\right]$. $V = 2\pi\left(2 - 1\right)$.

Answer:

$2\pi$