Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis.
-y = 3x, y = 6x, x = 3
A) $54 \pi$
B) $18 \pi$
C) $\frac { 243 } { 2 } \pi$
D) $27 \pi$

Question

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis.
-y = 3x, y = 6x, x = 3 
A) $54 \pi$
B) $18 \pi$
C) $\frac { 243 } { 2 } \pi$
D) $27 \pi$

Quizplus · Accepted Answer

The volume of the solid generated by revolving the region bounded by the curves y = 3x, y = 6x, and the line x = 3 about the y-axis can be found using the shell method. The formula for the shell method is $V = 2\pi \int_{a}^{b} x \cdot f(x) dx$, where $f(x)$ is the height of the shell and $x$ is the radius of the shell. In this case, the height of the shell is the difference between the two functions, $6x - 3x = 3x$, and the radius of the shell is $x$. The limits of integration are from 0 to 3 (the bounds for $x$). Thus, the volume is $V = 2\pi \int_{0}^{3} x(3x) dx = 2\pi \int_{0}^{3} 3x^2 dx = 2\pi [x^3]_{0}^{3} = 2\pi [27 - 0] = 54\pi$.

Use the Shell Method to Find the Volume of the Solid