Question

calcular a integral tripla 
F(x,y,z) = x na ordem dzdydx
x( 0 , 1)
y(0,1-x)
z(0,1-x-y)

Answer 1

Olá, Bruno.

$\int_0^{1}\int_0^{1-x}\int_0^{1-x-y} x\,dz\,dy\,dx=\int_0^{1}\int_0^{1-x}x(1-x-y)\,dy\,dx=\\\\ =\int_0^{1}\int_0^{1-x}(x-x^2-xy)\,dy\,dx=\int_0^{1}(xy|_0^{1-x}-x^2y|_0^{1-x}-x\frac{y^2}2|_0^{1-x})\,dx=\\\\ =\int_0^{1}[x(1-x)-x^2(1-x)-\frac x 2(1-x)^2]\,dx=\\\\ =\int_0^{1}[x-x^2-x^2+x^3-\frac x 2(1-2x+x^2)]\,dx=\\\\ =\int_0^{1}[x-2x^2+x^3-\frac x 2+x^2-\frac{x^3}2]\,dx=\\\\ =\int_0^{1}(\frac x 2-x^2+\frac{x^3}3)\,dx=\\\\ =\frac{x^2}4|_0^1-\frac{x^3}{3}|_0^1+\frac{x^4}{12}|_0^1=$

$=\frac14-\frac13+\frac{1}{12}=\frac{3-4+1}{12}=\\\\ =\boxed{0}$