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

Explanation:

Step1: Recall the disk - method formula

The volume $V$ of the solid of revolution about the $x$-axis using the disk - method for a function $y = f(x)$ on the interval $[a,\infty)$ is given by $V=\pi\int_{a}^{\infty}[f(x)]^{2}dx$. Here, $a = 2$ and $f(x)=x^{-6}$, so $V=\pi\int_{2}^{\infty}(x^{-6})^{2}dx=\pi\int_{2}^{\infty}x^{- 12}dx$.

Step2: Evaluate the improper integral

The improper integral $\int_{2}^{\infty}x^{-12}dx=\lim_{b\rightarrow\infty}\int_{2}^{b}x^{-12}dx$. Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $\lim_{b\rightarrow\infty}\int_{2}^{b}x^{-12}dx=\lim_{b\rightarrow\infty}\left[\frac{x^{-12 + 1}}{-12+1}\right]{2}^{b}=\lim{b\rightarrow\infty}\left[-\frac{1}{11x^{11}}\right]_{2}^{b}$.

Step3: Evaluate the limit

$\lim_{b\rightarrow\infty}\left(-\frac{1}{11b^{11}}+\frac{1}{11\times2^{11}}\right)$. As $b\rightarrow\infty$, $\frac{1}{11b^{11}}\rightarrow0$. So $\lim_{b\rightarrow\infty}\left(-\frac{1}{11b^{11}}+\frac{1}{11\times2^{11}}\right)=\frac{1}{11\times2^{11}}$.

Step4: Calculate the volume

$V=\pi\times\frac{1}{11\times2^{11}}=\frac{\pi}{11\times2048}=\frac{\pi}{22528}$.

Answer:

A. The volume is $\frac{\pi}{22528}$ cubic units.