   Chapter 4.1, Problem 8E

Chapter
Section
Textbook Problem

# Evaluate the upper and lower sums for f ( x ) = 1 + x 2 ,   − 1 ≤ x ≤ 1 , with n = 3 and 4. Illustrate with diagrams like Figure 14.

To determine

To evaluate:

The upper and lower sums for fx=1+x2,  -1x1

Explanation

1) Concept:

i) Width of the interval [a, b] is (b-a)

ii) Width of each strip is x=b-an

iii) Upper sum: Un=fx1x+fx2x+..+fxnx

iv) Lower sum: Ln=fx1x+fx2x+..+fxnx

2) Given:

fx=1+x2,  -1x1

3) Calculation:

x=1-(-1)n=2n

The graph of the function is

i) Divide the given area into 3 rectangle strips by drawing vertical lines at  x=-13, 13

Graph

For n=3  x=23

The maximum value occurs at the end points of the interval, i.e., at x=-1 ,x=1/3, x=1

Therefore, the upper sum is

=f-1·23+f13·23+f1·23

=2·23+109·23+2·23=9227

U33.41

The minimum values occurs at x=-13,  x=0,  x=13

Therefore, the lower sum is

=f-13·23+f0·23+f13·23

=109·23+1·23+109·23=5827

L32

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