第8章 8.9

$\iiint_D x^2 y\,dxdydz \qquad D: x \geq 0 \quad y \geq 0 \quad z \geq 0 \quad x+y+z \leq 1$

$\int_0^1\int_0^{1-x}\int_0^{1-x-y} x^2 y\,dzdydx = \int_0^1\int_0^{1-x} \left[x^2yz\right]_0^{1-x-y} dydx = \int_0^1\int_0^{1-x} x^2y - x^3y - x^2y^2\,dydx$

$= \int_0^1\left[\frac{1}{2}x^2y^2 - \frac{1}{2}x^3y^2 - \frac{1}{3}x^2y^3\right]_0^{1-x} dx = \int_0^1 \frac{1}{2}x^2(1-x)^2 - \frac{1}{2}x^3(1-x)^2 - \frac{1}{3}x^2(1-x)^3\,dx$

$= \int_0^1 \frac{1}{2}x^2(1-3x+3x^2-x^3) - \frac{1}{3}x^2(1-3x+3x^2-x^3)\,dx$

$= \left[\frac{1}{6}x^3 - \frac{3}{8}x^4 + \frac{3}{10}x^5 - \frac{1}{12}x^6 - \frac{1}{9}x^3 + \frac{1}{4}x^4 - \frac{1}{5}x^5 + \frac{1}{18}x^6\right]_0^1 = \frac{1}{360}$