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.
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.
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