[12.1] - Double integrals from data

Some of the values of the function $f(x,y)$ are given in the table below:

Estimate $$\int_{y=80}^{100}\int_{x=20}^{60} f(x,y)\,dx\,dy\approx\sum_{i=1}^2\sum_{j=1}^2 f(x_{ij}^*,y_{ij}^*)\Delta x\Delta y$$ where $\Delta x=20$ and $\Delta y=10$. (That is, the 2 X 2 light overlay). Which point to use in each sub-rectangle? Try:

1. midpoints
   
   
   
   
   
   
   
   
   
2. points farthest away from the origin