(837, #9) Let V be the volume that lies below f(x,y) = sqrt(52 - x² - y²) and above the rectangle R = [2,4] x [2,6] in the xy-plane. Partion R by means of the lines x = 3 and y = 4. Let L be the Riemann sum computed by using the lower left corner of the partitioning rectangles, and U be the one corresponding to the upper right corners. Without calculating the numbers V, L, and U, arrange them in increasing order and explain your reasoning.

Solution: The surface of the function f(x,y) lies on the sphere x² + y² + z² = 52 . Consequently, see the graph, the lower, left corner of each subrectangle will coorespond to the maximum of the function on the subrectangle. Thus we will have L >= V.

Similarly, the upper, right corners correspomnd to minima, so V >= U.

by:rjm