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The region between the graphs of x=y^2 and x=6y is rotated around the line y=6. The volume of the resulting solid is ____?Asked by RaShawnya
The region between the graphs of y=x^2 and y=6x is rotated around the line x=8. The volume of the resulting solid is?
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Answered by
Steve
The curves intersect at (0,0) and (6,36). So, using discs (washers) of thickness dy,
v = ∫[0,36] π(R^2-r^2) dy
where R = 8-y/6 and r = 8-√y
v = ∫[0,36] π((8-y/6)^2-(8-√y)^2) dy = 360π
using shells of thickness dx,
v = ∫[0,6] 2πrh dx
where r=8-x and h=6x-x^2
v = ∫[0,6] 2π(8-x)(6x-x^2) dx = 360π
v = ∫[0,36] π(R^2-r^2) dy
where R = 8-y/6 and r = 8-√y
v = ∫[0,36] π((8-y/6)^2-(8-√y)^2) dy = 360π
using shells of thickness dx,
v = ∫[0,6] 2πrh dx
where r=8-x and h=6x-x^2
v = ∫[0,6] 2π(8-x)(6x-x^2) dx = 360π