32. find the volume of the solid generated by revolving the region bounded by the curve $y =…

Explanation:

Step1: Find the intersection point

Set $y = 1$ in $y=\sqrt{\frac{x}{2}}$, then $1=\sqrt{\frac{x}{2}}$. Squaring both sides gives $x = 2$.

Step2: Use the disk - method formula

The formula for the volume $V$ of the solid of revolution about the $x$-axis using the disk method is $V=\pi\int_{a}^{b}[f(x)]^{2}dx$. Here, $a = 0$, $b = 2$ and $f(x)=\sqrt{\frac{x}{2}}$, so $V=\pi\int_{0}^{2}(\sqrt{\frac{x}{2}})^{2}dx$.

Step3: Simplify the integrand

$(\sqrt{\frac{x}{2}})^{2}=\frac{x}{2}$, so $V=\pi\int_{0}^{2}\frac{x}{2}dx$.

Step4: Integrate

$\int\frac{x}{2}dx=\frac{1}{2}\times\frac{x^{2}}{2}+C=\frac{x^{2}}{4}+C$. Then $V=\pi\left[\frac{x^{2}}{4}\right]_{0}^{2}$.

Step5: Evaluate the definite - integral

$V=\pi\left(\frac{2^{2}}{4}-\frac{0^{2}}{4}\right)=\pi(1 - 0)=\pi$.

Answer:

$\pi$