- #1

zeus77

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## Homework Statement

Volume enclosed by: z= x^2 + y^2 and z= 2y.

This should look like 1/2 of a bowl only on the +y side of the x y z plane I believe. I want to know if integrating in different order or cord system will affect the result. I am having a lot of problems attempting to find the answer to this part of the equation:

V= Int(0->2) [4/3 (2y-y^2)^(3/2) dy] = Pi/2 ?

My teacher says he found the volume to be Pi/2. Which i think might be incorrect.

I need to show the process of integration of dydzdx and dxdydz and compare the results. Or change to spherical coordinates.

**2. The attempt at a solution**

I am having an horrendous time finding the correct bounds for this i believe is my problem. Along with the fact i do not understand how to use my accessible integration tables to correctly solve the last part.

Starting with:

V= Int[0->2]dy Int[y^2 -> 2y]dz Int [-sqrt(z-y^2) -> sqrt(z-y^2)]

I move down correctly to,

V= Int(0->2) [4/3 (2y-y^2)^(3/2) dy]

V= Pi/2 ?

Can some one please explain to me the correct integration table substitution of this part? Every table i look at has things in the form of:

Int[sqrt(2au-u^2)du , Int[sqrt(u^2-a^2) or Int[sqrt(a^2-u^2)

How do you convert my problem which seems to have the form of Int[{(sqrt(2au-u^2))^3}du] ?? I've tried of substitution and i feel uncertain of the result. If someone could explain how that integral equals Pi/2 i should be able to change to spherical cords or integrate in a different order.

Thanks in advance!

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