Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method.
-The region bounded by $y = 3 x - x ^ { 2 }$ and $y = x$ about the $y$-axis
A) $\frac { 8 } { 3 } \pi$
B) $4 \pi$
C) $2 \pi$
D) $\frac { 4 } { 3 } \pi$

Question

Quizplus · Accepted Answer

To find the volume of the solid generated by revolving the region bounded by $y = 3x - x^2$ and $y = x$ about the $y$-axis, we can use the shell method. First, we need to find the points of intersection between the two curves, which occurs when $3x - x^2 = x$. Simplifying this, we get $x^2 - 2x = 0$, leading to $x( x - 2) = 0$, so the points of intersection are $x = 0$ and $x = 2$.The volume $V$ of the solid can be calculated using the formula for the shell method: $$V = 2\pi \int_{a}^{b} x(f(x) - g(x)) dx$$, where $f(x) = 3x - x^2$ and $g(x) = x$, and the limits of integration are from $a = 0$ to $b = 2$.Substituting the given functions and limits into the formula gives: $$V = 2\pi \int_{0}^{2} x((3x - x^2) - x) dx = 2\pi \int_{0}^{2} x(2x - x^2) dx = 2\pi \int_{0}^{2} (2x^2 - x^3) dx$$.Integrating, we get: $$V = 2\pi \left[ \frac{2}{3}x^3 - \frac{1}{4}x^4 \right]_{0}^{2} = 2\pi \left[ \frac{2}{3}(8) - \frac{1}{4}(16) \right] = 2\pi \left[ \frac{16}{3} - 4 \right] = 2\pi \left[ \frac{8}{3} \right] = \frac{16}{3}\pi$$.However, there seems to be a mistake in my calculation. Correctly integrating and evaluating gives: $$V = 2\pi \left[ \frac{2}{3}x^3 - \frac{1}{4}x^4 \right]_{0}^{2} = 2\pi \left[ \frac{2}{3}\cdot8 - \frac{1}{4}\cdot16 \right] = 2\pi \left[ \frac{16}{3} - 4 \right] = 2\pi \left[ \frac{16}{3} - \frac{12}{3} \right] = 2\pi \left[ \frac{4}{3} \right]$$.This indicates a miscalculation in my final step. The correct evaluation should indeed reflect the formula and integration process accurately. Given the setup and the method, the correct volume should be derived from the integral setup and calculation. The correct answer, based on the correct integration and evaluation process, should be $\frac{8}{3}\pi$, which corresponds to choice A, acknowledging the error in my explanation steps.

Find the Volume of the Solid Generated by Revolving the Region