I| (+ y-52) Evaluate dV where E E = {(x, y, z) | – 4< y < 0,0

Let $V$ be the value of given triple integral is:

$V=\int \int {\int }_E \left(x+y-5z\right) dV$

where $E=\left\{\left(x, y, z\right) | -4\le y\le 0, 0\le x\le y, 0<z<x+y^2\right\}$.

Notice that value of $z$ changes from $0$ to $x+y^2$. Substitute the limits for $z$:

$V=\int \int {\int }_0^{x+y^2} \left(x+y-5z\right) dz ....$

The value of $x$ changes from $0$ to $y$. Substitute the limits for $x$:

$V=\int {\int }_0^y{\int }_0^{x+y^2} \left(x+y-5z\right) dz dx ..$

The value of $y$ changes from $-4$ to $0$. Substitute the limits for $y$:

$V={\int }_{-4}^0{\int }_0^y{\int }_0^{x+y^2} \left(x+y-5z\right) dz dx dy$