volumes - Math 221-16 Calculus I Section 6.2(Volumes Defn...

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Math 221-16: Calculus ISection 6.2 (Volumes)Defn of Volume: LetSbe a solid that lies betweenx=aandx=b. If the cross-sectional area ofSin the planePxthroughxand perpendicular to thex-axis, isA(x)whereAis a continuous function, then the volume ofSisV= limn→∞nsummationdisplayi=1A(x*ix=integraldisplaybaA(x)dxExamples (Solids of Revolution)1. Find the volume of the solid obtained by rotating the region bounded byy=x3,y= 8 andx= 0 about they-axis.2. The region enclosed byy=xandy=x2is rotated about thex-axis. Find thevolume of the resulting solid.3. Find the volume of the solid obtained by rotating the region enclosed byy=xandy=x3about the linex=-14. Find the volume of a solid obtained by rotating about thexaxis the regionunder the curvey=xbetween from 0 to 1.5. Find the volume of the solid obtained by rotating the region enclosed byy=xandy=xabout the liney= 1.6. The region bounded byy=r,x= 0,x=h, and thex-axis is rotated aroundthex-axis. Find its volume.

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