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Dilemma

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Hello everyone,

Question : Find the volume of the region which remains inside the cyclinder x 2 + y 2 = 2y, and is bounded from above by the paraboloid surface x 2 + y 2 + z = 1 and from below by the plane z = 0

This looks like a pretty straightforward question. First thing to do is converting equations to polar coordinates. When they are converted, the cylinder's region can be written as r = 2sinθ. Parabolaid surface's eq. is (1-r

∫(from 0 to π)∫from 0 to 2sinθ) (1-r

However, the solution to this problem proposes a different approach which does not return the same answer.

Here it is:

Thanks in advance,

1. Homework Statement1. Homework Statement

Question : Find the volume of the region which remains inside the cyclinder x 2 + y 2 = 2y, and is bounded from above by the paraboloid surface x 2 + y 2 + z = 1 and from below by the plane z = 0

## Homework Equations

## The Attempt at a Solution

This looks like a pretty straightforward question. First thing to do is converting equations to polar coordinates. When they are converted, the cylinder's region can be written as r = 2sinθ. Parabolaid surface's eq. is (1-r

^{2}).Through these, a double integral can easily be set up as the following:∫(from 0 to π)∫from 0 to 2sinθ) (1-r

^{2})rdrdθ.However, the solution to this problem proposes a different approach which does not return the same answer.

Here it is:

Thanks in advance,

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