Given the equation: $$Z= 2 x^3 +6 y^2 x -3 y^3 - 150 x.$$ Find the volume in surface $Z$ for $z=0$, $x=8$, $y=0$,$y=\frac{x}{2}$ using an arithmetic (approximate) solution.
I understand it can be done with double integral of $Z$ with the limits of $x = 0$ to $8$ and $y = 0$ to $x/2$, but explicitly asks for arithmetic means. We are given a table of data, which may help but I am unsure what to do.
Thanks in advance
It seems odd that you would be asked to use a numerical integration when that polynomial function seems relatively easy to integrate. However, since you are told to do this numerically, the x and y values differ by 1 so $\Delta x$ and $\Delta y$ are 1. You multiply each z value by 1 which means, of course, that you use the z-values themselves. Since y goes from 0 to x/2, you add the values of z in the table inside the triangle with x going from 0 to 8 and y from 0 to x/2. That triangle has vertices at (x, y)= (0, 0), z= 0; (x, y)= (0, 4), z= -192; (x, y)= (8, 0), z= 400.