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Video Example) Tutorial Online Textbook EXAMPLE 4 Find the volume of the solid that lies under the paraboloid z = 3x2 + 3y2, above the xy-plane, and inside the cylinder x2 + y2 = 8x. SOLUTION The solid lies above the disk D whose boundary circle has equation x2 + y2 = 8x or, after completing the square, (x - 1 x)2 + y2 = 1 In polar coordinates we have x2 + y2 = rand x = rcos(6), so the boundary circle becomes r = ,or r = 8cos(). Thus the disk D is given by D = {(r, 0) 1 -1/2 < 1/2,0 Srs 8cos(O)} PT/2 8 cos(0) V= -1/2 JO L2%** r dr do /2 1800 (6) JO do = 3072 (cos())*de /2 (cos())*de /2 1 + cos(20) de 2 [1 + 2cos(20) + 12(1 + cos(40))]de = 1536 16/2

Answer 1

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