Find the volume of the solid whose base is the region in the​ xy-plane that is bounded by the parabola (in Des.) and the line (in Des.)​, while the top of the solid is bounded by the plane (in Des.).

Question

Find the volume of the solid whose base is the region in the​ xy-plane that is bounded by the parabola y = 4 − x² and the line y = −3x​, while the top of the solid is bounded by the plane z = x+6.

Answer 1

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)∫_-3x^{4-x^2}dydx = ∫_-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