Question

Find the volume of the solid whose base is the region in the xy-plane that is bounded by the parabola y=4-x^2 and the line y=-3x, while the top of the solid is bounded by the plane z=x+5

Answer 1

Step-by-step explanation:

Limits of integration: 4 - x2 = - 3x; x2 - 3x - 4 = 0; (x - 4)(x + 1) = 0; x = -1 or x = 4;

V = ∫∫R(x +6)dxdy = ∫-14(x + 6)∫-3x4-x^2dydx = ∫-14(x + 6)(4 - x2 + 3x)dx = ∫-14(4x-x3 +3x2+24 - 6x2 + 18x)dx =

∫-14(22x-x3-3x2+24)dx = (11x2-x4/4 - x3+24x)-14= (176 - 64 - 64 + 96) - (11-1/4 + 1 - 24) = 156.25