Find the volume of the solid that lies under the paraboloid z = 8x2 + 8y2, above the xy-plane, and inside the cylinder x2 + y2 = 2x.

Given the paraboloid z = 8x2+8y2 , above the xy-plane, and inside the cylinder x2+y2 = 2x.Now Volume V=∫∫DzdAV=∫∫D8x2+8y2dA ....1Now by change in the polar coordinatewe know that x=rcosθ , y=rsinθ givesx2+y2=r2Thenx2+y2 = 2xr2=2rcosθr=2cosθThen the limit is given 0≤r≤2cosθ , -π2≤θ≤π2Now V=∫θ=-π2π2∫r=02cosθ8r2rdrdθV=2∫θ=0π2∫r=02cosθ8r3drdθ V=16∫θ=0π2∫r=02cosθr3drdθV=16∫θ=0π2r44r=02cosθdθV=4∫θ=0π22cosθ4-0dθV=64∫θ=0π2cos4θdθV=64∫θ=0π2cos4θdθthus, 2cos2θ=1+cos2θcos2θ=1+cos2θ2cos4θ=1+cos2θ22cos4θ=1+cos22θ+2cos2θ4NowV=64∫θ=0π21+cos22θ+2cos2θ4dθV=16∫θ=0π21+cos22θ+2cos2θdθV=16∫θ=0π21+1+cos4θ2+2cos2θdθV=16θ+12θ+sin4θ4+sin2θθ=0π2V=16π2+12π2+sin2π4+sinπ-0V=16π2+12π2+0+0 because sin2π=0 ,sinπ=0, sin0=0 V=16π2+π4V=163π4V=12π

Answered: Find the volume of the solid that lies…

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter10: Analytic Geometry

Section10.1: The Rectangular Coordinate System

Problem 38E: Find the exact volume of the solid that results when the triangular region with vertices at 0, 0, 6,...

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Find the volume of the solid that lies under the paraboloid

z = 8x² + 8y², above the xy-plane, and inside the cylinder

x² + y² = 2x.

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