Find the volume of the solid obtained by rotating the region bounded by the given curves

Mathematics
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Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. 

y=x2x=y2 about the axis x=5 
May 1st, 2015

The domain bounded by the curves y = x2 and x = y2 can be described by the inequalities:

0 ≤ y ≤ 1, y2 ≤ x ≤ √(y). Then the volume can be expressed as the following integral:

V = π ∫01 [x22(y) – x12(y)] dy, where x2(y) = √(y) + 5 and x1(y) = y2 + 5.

Thus, V = π ∫01 [(√(y) + 5)2 – ( y2 + 5)2] dy =

π ∫01 [(y + 10√(y) + 25) – (y4 + 10y2 + 25)] dy =

π ∫01 [y + 10√(y) – y4 – 10y2 ] dy =

π ( y2/2 + (20/3)y√(y) – y5/5 – (10/3)y3 )]01 =

π ( 1/2 + 20/3 – 1/5 – 10/3) = π ( 1/2 + 10/3 – 1/5) = π ( 15 + 100 – 6)/30 = 109π/30.


May 1st, 2015

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