Asked by mrbeas
Use the washer method to determine the volume of the solid formed when the region bounded by y=1/2x^2, y=0, and x=2 is rotated about the line y=3.
Answers
Answered by
oobleck
The curves intersect at (2,2)
So, using washers of thickness dx
v = ∫[0,2] π(R^2-r^2) dx
where R = 3-y = 3 - 1/2 x^2 and r= 3-2 = 1
v = ∫[0,2] π((3 - 1/2 x^2)^2-1^2) dx = 48π/5
to check your answer, use shells of thickness dy
v = ∫[0,2] 2πrh dy
where r=3-y and h = x = √(2y)
v = ∫[0,2] 2π(3-y)√(2y) dy = 48π/5
So, using washers of thickness dx
v = ∫[0,2] π(R^2-r^2) dx
where R = 3-y = 3 - 1/2 x^2 and r= 3-2 = 1
v = ∫[0,2] π((3 - 1/2 x^2)^2-1^2) dx = 48π/5
to check your answer, use shells of thickness dy
v = ∫[0,2] 2πrh dy
where r=3-y and h = x = √(2y)
v = ∫[0,2] 2π(3-y)√(2y) dy = 48π/5
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