   Chapter 15.1, Problem 1E

Chapter
Section
Textbook Problem

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

(a)

To determine

To estimate: The volume of the solid lies below the surface and above the rectangle by taking the sample points in the upper right corner of each square.

Explanation

Formula used:

The volume of the given solid is, Vlimm,ni=1mj=1nf(xi,yj)ΔA ,

Here, ΔA=a2 where a is the length of the side of the square.

The sample points of the upper right corner of each square is denoted as (xi,yj) .

The image value of the sample points under the function f is denoted as f(xi,yj) .

And, m, n is the Riemann sum constants.

Given:

The surface, z=f(x,y)=xy .

The rectangle, R={(x,y)|0x6,0y4} .

The Riemann sum constants, m=3,n=2 .

Calculation:

Plot the given rectangle in the graph and pick the sample points at the upper right corners of each square.

From Figure 1, it is observed that the upper right corners of each square is (2,2),(2,4),(4,2),(4,4),(6,2),(6,4) and a=2 .

Therefore, ΔA=4 .

Substitute 2 for x and 2 for y in the given function z=f(x,y)=xy .

f(2,2)=2×2=4

Substitute 2 for x and 4 for y in the given function z=f(x,y)=xy .

f(2,4)=2×4=8

Substitute 4 for x and 2 for y in the given function z=f(x,y)=xy

(b)

To determine

To estimate: The volume of the solid lies below the surface and above the rectangle by using Midpoint Rule.

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