Volume of given solid

**purplerain** · October 16th 2009, 07:16 PM

Find the volume of the given solid.
Bounded by the cylinder y^2 + z^2 = 4 and the planes x = 2y, x = 0, z = 0 in the first octant

**HallsofIvy** · October 17th 2009, 02:48 AM

Originally Posted by purplerain

Find the volume of the given solid.
Bounded by the cylinder y^2 + z^2 = 4 and the planes x = 2y, x = 0, z = 0 in the first octant

Because of the obvious cylindrical symmetry, I would recommend changing to polar coordinates in the yz-plane. Then $y= r cos(\theta)$ , $z= r sin(\theta)$ and the cylinder, $y^2+ z^2= 4$ becomes r= 2 and x= 2y gives height $x= 2r cos(\theta)$ .
Integrate $2r cos(\theta) r dr d\theta$ with r ranging from 0 to 2 and $\theta$ ranging from 0 to $\pi/2$ .

(Normally, of course, we use polar coordinates in an xy-plane. Essentially what I did was swap x and z so the problem would be "find the area of the solid bounded by $x^2+ y^2= 4$ in the first quadrant from z= 0 to z= 2y. Changing the coordinate system doesn't change the solid so the two solids have the same volume)

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