Question

Find the volume of the solid in R^3 bounded by y=x^2, x=y^2, z=x+y+15, and z=0.

Answer 1

$V = \iiint dV = \int_0^1 \int_{x^2}^{\sqrt x} \int_0^{x+y+15}dzdydx \\ = \int_0^1 \int_{x^2}^{\sqrt x} x + y+15 dydx \\ = \int_0^1 \Big[xy + 1/2y^2 + 15y\Big]_{x^2}^{\sqrt x} dx \\ = \int_0^1 x\sqrt x + 1/2x+ 15\sqrt x - x^3 - 1/2x^4 - 15x^2 dx \\ = 2/5x^{5/2} + 1/4 x^2 + 10x^{3/2} - 1/4x^4 - 1/10x^5 - 5x^3\Big|_0^1 \\ = 2/5 + 1/4 + 10 - 1/4 - 1/10 - 5 = 5.3$