find the volume of the described solid of revolution or state that it does not exist. the region bounded by…

Explanation:

Step1: Recall volume - of - revolution 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, $a = 0$, $b=\frac{\pi}{2}$, and $f(x)=- 10\tan(x)$. So, $V=\pi\int_{0}^{\frac{\pi}{2}}(-10\tan(x))^{2}dx=\pi\int_{0}^{\frac{\pi}{2}}100\tan^{2}(x)dx$.

Step2: Use trigonometric identity

Recall the identity $\tan^{2}(x)=\sec^{2}(x)-1$. Then the integral becomes $V = 100\pi\int_{0}^{\frac{\pi}{2}}(\sec^{2}(x)-1)dx$.

Step3: Integrate term - by - term

We know that $\int\sec^{2}(x)dx=\tan(x)$ and $\int 1dx=x$. So, $V = 100\pi\left[\tan(x)-x\right]_{0}^{\frac{\pi}{2}}$.

Step4: Evaluate the definite integral

As $x\rightarrow\frac{\pi}{2}^{-}$, $\tan(x)\rightarrow+\infty$. So, $100\pi\left[\tan(x)-x\right]{0}^{\frac{\pi}{2}}=100\pi\left(\lim{x\rightarrow\frac{\pi}{2}^{-}}\tan(x)-\frac{\pi}{2}-(0 - 0)\right)=+\infty$.

Answer:

B. The volume does not exist.