find the volume of the solid generated by revolving the region bounded by the given lines and curves about…

Explanation:

Step1: 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), where (f(x)) is the function, and ([a,b]) is the interval of integration. Here, (f(x)=\sqrt{x}), (a = 0), and (b=9). So, (V=\pi\int_{0}^{9}(\sqrt{x})^{2}dx).

Step2: Simplify the integrand

Since ((\sqrt{x})^{2}=x) for (x\geq0), the integral becomes (V=\pi\int_{0}^{9}x dx).

Step3: Integrate

Using the power - rule (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C) ((n\neq - 1)), for (n = 1), (\int xdx=\frac{x^{2}}{2}+C). Then (V=\pi\left[\frac{x^{2}}{2}\right]_{0}^{9}).

Step4: Evaluate the definite integral

(V=\pi\left(\frac{9^{2}}{2}-\frac{0^{2}}{2}\right)). (V=\pi\times\frac{81}{2}).

Answer:

(\frac{81}{2}\pi) (corresponding to the fourth option)