Find the volume of the solid that lies under the paraboloid z=4x2+4y2, above the xy-plane, and inside the cylinder x2+y2

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Find The Volume Of The Solid That Lies Under The Paraboloid Z 4x2 4y2 Above The Xy Plane And Inside The Cylinder X2 Y2 1 (69.96 KiB) Viewed 38 times

Find the volume of the solid that lies under the paraboloid z=4x2+4y2, above the xy-plane, and inside the cylinder x2+y2=2x. Solution The solid lies above the disk D whose boundary circle has equation x2+y2=2x or, after completing the square, (x−)2+y2= In polar coordinates we have x2+y2=r2 and x=rcos(θ), so the boundary circle becomes r2=2rcos(θ), or r=2 cos (θ). Thus the disk D is given by D={(r,θ)∣−π/2≤θ≤π/2,0≤r≤2cos(θ)} and, by this formula, we have the following. V=∬D​(4x2+4y2)dA =∫−π/2π/2​∫02cos(θ)​(1)rdrdθ =∫−π/2π/2​[]02cos(θ)​dθ =∫0π/2​cos4(θ)dθ =∫0π/2​(21+cos(2θ)​)2dθ =∫0π/2​[1+2cos(2θ)+21​(1+cos(4θ))]dθ