Asked by ANON
A region is bounded by y=2x and y=4sqrt(x). Find the volume when rotated along
1)y=-1
2)the y axis
1)y=-1
2)the y axis
Answers
Answered by
Steve
cmon, guy - these are all basically the same. Figure the boundary, decide whether to use shells or discs, and add 'em up. For this one, either will work quite well.
cylinders of thickness dx (or dy) have a volume of 2πrh dx
discs have a volume of πr^2 dx
the curves intersect at (0,0) and (4,8)
#1
shells: v = ∫[0,8] 2π(y+1)(y/2 - y^2/16) dy
discs: v = ∫[0,4] π((4√x+1)^2 - (2x+1)^2) dx
#2
shells: v = ∫[0,4] 2πx(4√x - 2x) dx
discs: v = ∫[0,8] π((y/2)^2-(y^2/16)^2) dy
Doing the volume both ways lets you check your answers to be sure they agree.
cylinders of thickness dx (or dy) have a volume of 2πrh dx
discs have a volume of πr^2 dx
the curves intersect at (0,0) and (4,8)
#1
shells: v = ∫[0,8] 2π(y+1)(y/2 - y^2/16) dy
discs: v = ∫[0,4] π((4√x+1)^2 - (2x+1)^2) dx
#2
shells: v = ∫[0,4] 2πx(4√x - 2x) dx
discs: v = ∫[0,8] π((y/2)^2-(y^2/16)^2) dy
Doing the volume both ways lets you check your answers to be sure they agree.
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