Estimate the volume of the solid that lies below the surface and above the rectangle
.
Use a Riemann sum with and , and take the sample point to be the upper right corner of each subrectangle. Then use the Midpoint Rule to estimate the volume of the same solid.
Let be the volume of the solid that lies under the graph of
and above the rectangle given by and . Use the lines and to divide into subrectangles. Let and be the Riemann sums computed in lower left and upper right corners respectively. Without calculating numbers , , and , arrange them in increasing order and explain your reasoning.
The figure shows level curves of a function in the square . Use the Midpoint Rule with to estimate .
Evaluate the double integral
over .
If , show that
.