Question

Let V be the volume of the solid that lies under the graph of f(x,y)= (52 − x^2 − y^2)^1/2 and above the rectangle given by 4 ≤ x ≤ 6, 0 ≤ y ≤ 4. We use the lines x = 5 and y = 2 to divide R into subrectangles. Let L and U be the Riemann sums computed using lower left corners and upper right corners, respectively. Without calculating the numbers V, L, and U, arrange them in increasing order and explain your reasoning.

Answer 1

the surface is a half-sphere:

x^2+y^2+z^2 = 52

Since it is concave downward, it is clear that L < V < U

Think of the area under an arch. Left sums underestimate it, and right sums overestimate it.