let $r$ be the region enclosed by the line $y = 1$, the line $x = 4$ and the curve $y=sqrt{x}+1$. a solid is…

Explanation:

Step1: Use the disk - washer method formula

The formula for the volume $V$ of a solid of revolution about the line $y = k$ using the disk - washer method is $V=\pi\int_{a}^{b}[R(x)^2]dx$, where $R(x)$ is the radius of the cross - sectional disk. Here, $k = 1$, $a = 0$, $b = 4$, and $R(x)=(\sqrt{x}+1 - 1)=\sqrt{x}$.

Step2: Set up the integral

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

Step3: Integrate the function

Using the power rule for integration $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), for $n = 1$, we have $\int xdx=\frac{x^{2}}{2}+C$. Then $\pi\int_{0}^{4}x dx=\pi\left[\frac{x^{2}}{2}\right]_{0}^{4}$.

Step4: Evaluate the definite integral

$\pi\left(\frac{4^{2}}{2}-\frac{0^{2}}{2}\right)=\pi\left(\frac{16}{2}-0\right)=8\pi$.

Answer:

$8\pi$