I NEED HELP :(
- Thread starter reis
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mmm4444bot
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Hello reis -- welcome to the boards. Please post your attempt, so that tutors can see what you tried. Thanks!
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HallsofIvy
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\(\displaystyle x^2+ y^2= 2x\) is equivalent to \(\displaystyle x^2- 2x+ 1+ y^2= (x- 1)^2+ y^2= 1\). In the xy-plane that is a circle with center at (1, 0) and with radius 1. In three dimensions it is a cylinder with center along the line with parametric equations x= 1, y= 0, z= t. We can cover that circle by taking x from 0 to 2 and y from \(\displaystyle -\sqrt{1- (x-1)^2}= -\sqrt{2x-x^2}\) to \(\displaystyle \sqrt{1- (x-1)^2}= \sqrt{2x- x^2}\). For each (x, y) in the xy-plane, z goes from \(\displaystyle \sqrt{4- x^2- y^2}\) to \(\displaystyle \sqrt{4- x^2- y^2}\).
The volume is given by \(\displaystyle \int_{x= 0}^2\int_{-\sqrt{2x-x^2}}^{\sqrt{2x-x^2}}\int_{-\sqrt{4- x^2-y^2}}^{\sqrt{4- x^2- y^2}} dzdydx\).
The volume is given by \(\displaystyle \int_{x= 0}^2\int_{-\sqrt{2x-x^2}}^{\sqrt{2x-x^2}}\int_{-\sqrt{4- x^2-y^2}}^{\sqrt{4- x^2- y^2}} dzdydx\).