Question

Find the volume of the solid generated by revolving the given region about the given axis.
The region bounded by x²= 4y and y = x/2; about the y-axis.

Answer 1

Functions are y=0.25x and y=0.5x.
Intersection points are at x=0 and x=2.
Revolve about y-axis: use shell method;
\[V=\int\limits_{0}^{2}2\pi x \frac{ 1 }{ 4 }x^2dx-\int\limits_{0}^{2}2\pi x \frac{ 1 }{ 2 }xdx\]

Answer 2

Oops, switched both integrals: \[V=\int\limits_{0}^{2}2\pi x \frac{ 1 }{ 2 }xdx - \int\limits_{0}^{2}2 \pi x \frac{ 1 }{ 4 }x^2dx\](graph of y=0.5 x is above y=0.25x² between x=0 and x=2).

Answer 3

can i use ring method on this problem?

Answer 4

You can, but you have to rewrite the functions to x=f(y) and integrate from y=0 to y=1.

Answer 5

how to set up the integral using ring method?

Answer 6

I first sketch the curves