Question

Estimate the volume of the solid that lies below the surface z = xy and above the following rectangle. R = (x, y)|0 ≤ x ≤ 6, 4 ≤ y ≤ 8 Use a Riemann sum with m = 3, n = 2, and take the sample point to be the upper right corner of each square

Answer 1

The volume of the solid under the surface z=xy above the rectangle R can be estimated using a Riemann sum with 3 intervals in the x-direction and 2 in the y-direction, taking the function's value at the upper right corner of each subrectangle and summing the products of these values with the area of each subrectangle.

Step-by-step explanation:

To estimate the volume of the solid that lies below the surface z = xy and above the rectangle R defined by 0 ≤ x ≤ 6, 4 ≤ y ≤ 8, using a Riemann sum with m = 3 and n = 2, and taking the sample point to be the upper right corner of each rectangle, you can proceed as follows:

• Divide the x-interval [0, 6] into m = 3 subintervals of equal length, Δx = 2.
• Divide the y-interval [4, 8] into n = 2 subintervals of equal length, Δy = 2.
• For each subrectangle in R, calculate the value of the function z = xy at the upper right corner (xi, yj).
• Sum up all these values, each multiplied by the area of the subrectangle (Δx * Δy).

The Riemann sum is thus Σ(xy)ΔxΔy for all sample points (xi, yj).