Solve the problem.
-The region between the curve $y = \frac { 1 } { x ^ { 2 } }$ and the $x$-axis from $x = \frac { 1 } { 3 }$ to $x = 3$ is revolved about the $y$-axis to generate a solid. Find the volume of the solid.
A) $2 \pi \ln 3$
B) $4 \pi \ln 3$
C) $\pi \ln 3 - \pi$
D) $2 \pi \ln 3 - \pi$

Question

Quizplus · Accepted Answer

To use the method of cylindrical shells, we need to express the function in terms of x. We have $y=\frac{1}{x^2} \Rightarrow x^2 = \frac{1}{y}$. Using this expression, we can write the integral for the volume of the solid as:

$$V = 2\pi \int_{1/3}^{3} x \cdot \left(\frac{1}{x^2}\right) dx = 2\pi \int_{1/3}^{3} \frac{dx}{x}$$

Evaluating this integral gives:

$$V = 2\pi \big(\ln3 - \ln(1/3)\big) = 4\pi \ln3$$

Therefore, the answer is (B).

Solve the Problem y=1x2y = \frac { 1 } { x ^ { 2 } }y=x21​

Solve the Problem $y = \frac { 1 } { x ^ { 2 } }$