Question

Alguém pode me ajudar nessa solução?

Answer 1

El sólido V es un tetraedro, esto es

$V=\left\{(x,y,z): 0\leq x\leq 1\,,\, 0\leq y\leq 1-x\,,\, 0\leq z\leq 1-x-y\right\}\\ \\ \\ \displaystyle I=\iiint \limits_{V}x^2yzdV\\ \\ \\ I=\int_{0}^{1}\int_{0}^{1-x}\int_{0}^{1-x-y}x^2yz\;dz\,dy\,dx\\ \\ \\ I=\dfrac{1}{2}\int_{0}^{1}\int_{0}^{1-x}x^2y\left.z^2\right|_{z=0}^{z=1-x-y}\\ \\ \\ I=\dfrac{1}{2}\int_{0}^{1}\int_{0}^{1-x}x^2y(1-x-y)^2\;dy\,dx\\ \\ \\ I=-\dfrac{1}{24}\int_{0}^{1}\left.x^2(x + y - 1)^3(x - 3y - 1)\right|_{y=0}^{y=1-x}dx$

$\displaystyle I=\dfrac{1}{12}\int_{0}^{1}x^2(x-1)^4dx\\ \\ \\ I=\dfrac{1}{1260}$