Find $\iint_Ay\,dA$, where $A$ is defined by $z=x+y^2$,$0\le x\le 1$ and $0\le y\le2$

Question

Can anyone help me to find the $\iint_Ay\,dA $ where $A$ is defined by $z=x+y^2$ and $0\le x\le1$ and $0\le y\le2$

What I was thinking to do is to put $y=\sqrt{z-x}$ but I am not sure that I am doing right because I cannot find the limits of the integral.

Answer 1

Running the following code in Mathematica:

z[x_, y_] := x + y^2
dA = Sqrt[1 + D[z[x, y], x]^2 + D[z[x, y], y]^2];
Integrate[y*dA, {x, 0, 1}, {y, 0, 2}]

we can get the following results:

(13 Sqrt[2])/3