A contour map is shown for a function f\displaystyle {f} on the square R=[0,4]×[0,4]\displaystyle {R}={\left[{0},{4}\right]}\times{\left[{0},{4}\right]}

242419105510101919[Graphs generated by this script: initPicture(-.5,4.5,-.5,4.5);axes(2,2,1,1,1);circle([2,0],.707);circle([2,0],1.581);circle([4,3],.707);circle([0,3],.707);line([-1,0],[4,5]);line([0,5],[5,0]);ellipse([5,3],2,1.5);ellipse([-1,3],2,1.5);fontstroke='blue';fontfill='blue';text([2,.5],'19');text([2,1.8],'10');text([1.6,3.6],'5');text([2.4,3.6],'5');text([2.8,3],'10');text([1.2,3],'10');text([3.5,3],'19');text([.5,3],'19');]

Use the Midpoint Rule with m=n=4\displaystyle {m}={n}={4} to estimate the value for Rf(x,y)dA\displaystyle \int\int_{{R}}{f{{\left({x},{y}\right)}}}{d}{A}