Problem with finding volume using shells

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SUMMARY

The discussion centers on calculating the volume of a solid generated by revolving the region bounded by the curve y=(1/2)x² and the line y=2 about the y-axis. The integral setup for the volume is V= 2π ∫ from 0 to 2 of x(1/2)(x)² dx, yielding a total volume of 4π. The user attempts to find the radius of a hole that occupies one-fourth of this volume, leading to the equation 2π ∫ from 0 to r of (1/2)(x³) dx = π, which results in an incorrect value of 41/2. The discussion highlights the need for clarity in problem-solving and adherence to forum guidelines regarding homework queries.

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asadpasat
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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 41/2, which is not right as i checked
 
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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...
 
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