   Chapter 4.1, Problem 7E

Chapter
Section
Textbook Problem

# Evaluate the upper and lower sums for f ( x ) = 2 + sin x , 0 ≤ x ≤ π , with n = 2, 4, and 8. Illustrate with diagrams like Figure 14.

To determine

To evaluate:

The upper and lower sums for fx=2+sinx, 0xπ

Explanation

1) Concept:

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

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

iii) Area: Rn=fx1x+fx2x+..+fxnx

2) Given:

fx=2+sinx, 0xπ

3) Calculation:

i) Divide the given area into 2 rectangle strips by drawing vertical lines at  x=π/2

x=π-0n=πn

Graph

For n=2,x=π/2

The maximum value of function at both the intervals occurs at x=π/2

Therefore, the upper sum is

=fπ2·π2+fπ2·π2

=3π

U29.42

The minimum value of function at both the intervals occurs at x=0 and π

Therefore, the lower sum is

=f0·π2+fπ·π2

=π+π

L26.28

ii) Divide the given area into 4 rectangle strips by drawing vertical lines at

x=π4 and x=3π4

x=π-04=π4

For n=4 x=π/4

Upper sum =fπ4+fπ2+fπ2+f3π4π4

=2+122+2+1+2+1+2+122π4

=10+2π4

U48.96

The lower sum is

Lower sum =f0+fπ4+f3π4+fππ4

=2+2+122+2+122+2π4

=8+2π4

L47

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