Asked by Yo Shunng
Area under y = -x^2 + 1 between x = 0 and x = 1 is rotated around x-axis. Find the volume by shell method. (Please solve it using step by step solution).
Answers
Answered by
Steve
Each shell has thickness dy, so its volume is 2πrh dy. Add them all up and you have
v = ∫[0,1] 2πrh dy
where r=y and h=x=√(1-y)
v = ∫[0,1] 2πy√(1-y) dy = 8π/15
As a check, using discs,
v = ∫[0,1] πr^2 dx
where r=y=(1-x^2)
v = ∫[0,1] π(1-x^2)^2 dx = 8π/15
v = ∫[0,1] 2πrh dy
where r=y and h=x=√(1-y)
v = ∫[0,1] 2πy√(1-y) dy = 8π/15
As a check, using discs,
v = ∫[0,1] πr^2 dx
where r=y=(1-x^2)
v = ∫[0,1] π(1-x^2)^2 dx = 8π/15
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