Question

Calcular a integral tripla F(x,y,z) = x na ordem dzdydx
x( 0 , 1)
y(x²,x)
z(0,2-x-y)

Answer 1

$\displaystyle I=\int_0^1dx\int_{x^2}^xdy\int_{0}^{2-x-y}x\,dz\\ \\ I=\int_0^1dx\int_{x^2}^xx\left.(z)\right|_0^{2-x-y}dy \\ \\ I=\int_0^1dx\int_{x^2}^xx(2-x-y)dy\\ \\ I=\int_0^1dx\int_{x^2}^x2x-x^2-xy\;dy\\ \\ I=\int_0^1\left.\left(2xy-x^2y-\frac{xy^2}{2}\right)\right|_{y=x^2}^{y=x}dx$

$\displaystyle I=\int_0^1\left(2x^2-x^3-\frac{x^3}{2}\right)-\left(2x^3-x^4-\frac{x^5}{2}\right)dx\\ \\ I=\int_0^12x^2-\frac{7x^3}{2}-x^4+\frac{x^5}{2}\;dx\\ \\ I=\left.\left(\frac{2x^3}{3}-\frac{7x^4}{8}-\frac{x^5}{5}+\frac{x^6}{12}\right)\right|_0^1\\ \\ I=\frac{2}{3}-\frac{7}{8}-\frac{1}{5}+\frac{1}{12}\\ \\ \boxed{I=-\frac{61}{40}}$