Question #124216
(a) Find the volume of the solid generated by revolving the region bounded by the curves y = x2 and y = 4x−x2 about the line y = 6.
1
Expert's answer
2020-07-01T18:44:03-0400

V=V1V2=V = V1 - V2 = π02(6x2)2dxπ02(6(4xx2))2dx=\pi\int^2_0 (6 - x^2)^2 dx - \pi\int^2_0 (6 -(4x - x^2))^2 dx ==π02((6x2)2(6(4xx2))2)dx=π02(6x26+4xx2)(6x2+6= \pi\int^2_0 ((6 - x^2)^2 - (6 -(4x - x^2))^2) dx = \pi\int^2_0 (6 - x^2 - 6 + 4x - x^2)(6 - x^2 + 6 -

4x+x2)dx=π02(2x2+4x)(124x)dx=- 4x + x^2) dx = \pi\int^2_0 (-2x^2 + 4x)(12 - 4x)dx = =π02(8x316x224x2+48x)dx=π(2x4=\pi\int^2_0 (8x^3 - 16x^2 - 24x^2 + 48 x)dx = \pi (2x^4 - 40x3/3+24x2)02=40x^3/3 + 24x^2)|^2_0 =

=π(32320/3+96)=π(128320/3)== \pi (32 - 320/3 +96) = \pi (128 - 320/3) = (384320)π3=64π3\dfrac{(384-320)\pi}{3} = \dfrac{64\pi}{3}


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