   Chapter 15.1, Problem 4E

Chapter
Section
Textbook Problem

(a) Estimate the volume of the solid that lies below the surface z = 1 + x2 + 3y and above the rectangle R = [1, 2] × [0, 3]. Use a Riemann sum with m = n = 2 and choose the sample points to be lower left corners.(b) Use the Midpoint Rule to estimate the volume in part (a).

(a)

To determine

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

Explanation

Formula used:

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

Here, ΔA=lb , where l,b is the length and breadth of the rectangle.

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

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

The Riemann sum constants are denoted by m, n.

Given:

The surface, z=f(x,y)=1+x2+3y .

The rectangle, R=[1,2]×[0,3] .

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

Calculation:

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

From Figure 1, it is observed by the upper right corners of each square is (1,0),(1,32),(32,0),(32,32) and l=12,b=32 .

Therefore, ΔA=34 .

Substitute x=1,y=0 in the given function z=f(x,y)=1+x2+3y .

f(1,0)=1+(1)2+3(0)=2

Substitute x=1,y=32 in the given function z=f(x,y)=1+x2+3y

(b)

To determine

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

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