Consider the region between the graphs of $f ( x ) = x ^ { 2 }$ and $g ( x ) = 3 x$ on the interval [0, 3].
-Use disk/washer method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the vertical line x = 4.
A) 12.5 $\pi$
B) 20.5 $\pi$
C) 18.5 $\pi$
D) 22.5 $\pi$

Question

Quizplus · Accepted Answer

The volume of the solid obtained by revolving the region between the curves $y = x^2$ and $y = 3x$ around the line $x = 4$ can be found using the washer method. The outer radius $R$ is the distance from $x = 4$ to $y = 3x$, which is $4 - x$, and the inner radius $r$ is the distance from $x = 4$ to $y = x^2$, which is $4 - x^2$. The volume $V$ is given by the integral from 0 to 3 (the points of intersection) of $\pi(R^2 - r^2)$ dx, which simplifies to $\pi \int_{0}^{3} ((4-x)^2 - (4-x^2)^2) dx$. Solving this integral gives the volume as $22.5 \pi$.

Consider the Region Between the Graphs Of f(x)=x2f ( x ) = x ^ { 2 }f(x)=x2

Consider the Region Between the Graphs Of $f ( x ) = x ^ { 2 }$