18 Use double integration to compute the volume of the solid

1.8) Use double integration to compute the volume of the solid that lies under the
plane x + 2y - z = 0 and above the region bounded by y = x and y = x^4

Solution

sol take helo using the Shell Method: about x = 4 y = 5x - x^2 and y = x limits: 5x - x^2 = x 5x - x - x^2 = 0 4x - x^2 = 0 x * (4 - x) = 0 ---> x = 0 & 4 height ===> 5x - x^2 - x -----> 4x - x^2 radius ====> 4 - x 4 ? 2p * (4 - x) * (4x - x^2) dx 0 4 ? 2p * (16x - 4x^2 - 4x^2 + x^3) dx 0 4 ? 2p * (16x - 8x^2 + x^3) dx 0 . . . . . .. . . . . . . . .. . . . .. . . .. . . 4 2p * [ 8x^2 - (8/3) * x^3 + (1/4) * x^4 ] . . . . . .. . . . . . . . .. . . . .. . . .. . ..0 2p * [ 8 * ( 4^2 - 0^2 ) - (8/3) * ( 4^3 - 0^3 ) + (1/4) * ( 4^4 - 0^4 ) ] 2p * [ 8 * (16 - 0) - (8/3) * (64 - 0) + (1/4) * (256 - 0) ] 2p * [ 128 - (512/3) + 64 ] 2p * (64/3) (128p/3)