Problem with finding volume using shells

[asadpasat]

A solid is generated by revolving the region bounded by y=(1/2)x² and y= 2 about the y-axis.
So what I did is set up my integral V= 2pi (integral sign from 0 to 2) x(1/2)(x)² dx
When I solve i get 4pi, and then I have to find the radius of the hole which has one fourth of the volume. So the volume of the hole is pi ( the hole is centered around the axis of revolution which is y-axis). Then I set equal integral from 0 to r (the radius of the hole ) is 2pi(1/2)(x³)dx = pi. the answer i get is 4^1/2, which is not right as i checked

 

[Mark44]

This appears to be a homework problem or one in a textbook. @asadpasat, please start a new thread in the Homework and Coursework section (Calculus subsection). This forum section is for conceptual calculus questions, not for homework problems.
Thanks...