Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis. $y = 14 x - x ^ { 2 } , x = 0 , y = 49$
A) $V = 2 \pi \int _ { 0 } ^ { 7 } x \left( 49 - \left( 14 x - x ^ { 2 } \right) \right) d x = \frac { 2,401 } { 6 } \pi$
B)$V = 2 \pi \int _ { 0 } ^ { 7 } x \left( 98 - \left( 14 x - x ^ { 2 } \right) \right) d x = \frac { 2,401 } { 6 } \pi$
C)$V = 2 \pi \int _ { 0 } ^ { 7 } x \left( 196 - \left( 14 x - x ^ { 2 } \right) \right) d x = \frac { 2,401 } { 6 } \pi$
D)$V = 2 \pi \int _ { 0 } ^ { 7 } x \left( 196 - \left( 14 x - x ^ { 2 } \right) \right) d x = \frac { 2,401 } { 3 } \pi$
E)$V = 2 \pi \int _ { 0 } ^ { 7 } x \left( 49 - \left( 14 x - x ^ { 2 } \right) \right) d x = \frac { 2,401 } { 3 } \pi$

Question

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis. $y = 14 x - x ^ { 2 } , x = 0 , y = 49$
A) $V = 2 \pi \int _ { 0 } ^ { 7 } x \left( 49 - \left( 14 x - x ^ { 2 } \right) \right) d x = \frac { 2,401 } { 6 } \pi$
B)$V = 2 \pi \int _ { 0 } ^ { 7 } x \left( 98 - \left( 14 x - x ^ { 2 } \right) \right) d x = \frac { 2,401 } { 6 } \pi$
C)$V = 2 \pi \int _ { 0 } ^ { 7 } x \left( 196 - \left( 14 x - x ^ { 2 } \right) \right) d x = \frac { 2,401 } { 6 } \pi$ 
D)$V = 2 \pi \int _ { 0 } ^ { 7 } x \left( 196 - \left( 14 x - x ^ { 2 } \right) \right) d x = \frac { 2,401 } { 3 } \pi$
E)$V = 2 \pi \int _ { 0 } ^ { 7 } x \left( 49 - \left( 14 x - x ^ { 2 } \right) \right) d x = \frac { 2,401 } { 3 } \pi$

Quizplus · Accepted Answer

The volume of the solid generated by revolving the region about the y-axis using the shell method is correctly set up in option A. The integral $2 \pi \int _ { 0 } ^ { 7 } x \left( 49 - \left( 14 x - x ^ { 2 } \right) \right) d x$ correctly represents the volume of the solid, where $49 - (14x - x^2)$ gives the height of each cylindrical shell, $x$ is the radius of the shell, and the integral is evaluated from $x = 0$ to $x = 7$, which are the bounds of the region in the x-direction. The evaluation of this integral gives $\frac { 2,401 } { 6 } \pi$, which matches the result in option A.

Use the Shell Method to Set Up and Evaluate the Integral