   Chapter 4.1, Problem 1E

Chapter
Section
Textbook Problem

# (a) By reading values from the given graph of f, use five rectangles to find a lower estimate and an upper estimate for the area under the given graph of f from x   =   0 to x   = 10 . In each case sketch the rectangles that you use.(b) Find new estimates using ten rectangles in each case. To determine

a)

To find:

i) A lower estimate for the area under the given graph

ii) An upper estimate for the area under the given graph

Explanation

1) Concept:

i) Width of the interval [a, b] is b-a and  n=5

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

iii) Area: Rn=fx1x+fx2x+ +fxnx

That is, Rn=xfx1+fx2+ +fxn

iv) Area: Ln=fx0x+fx1x+ +fxn-1x

That is, Ln=xfx0+fx1+ +fxn-1

2) Given:

Graph of f from  x=0 to x=10, n=5

3) Calculation:

Note that the area of the given graph must lie between 11 and 50 because the graph contains the rectangle with side length 10 &5.

Let’s consider area as S.

Divide S into 5 strips S1, S2, S3, S4, and S5 by drawing vertical lines at

x=2, x=4, x=6, x=8

i) Lower estimate:

Graph (a):

To determine

b)

To find:

i) A lower estimate for the area under the given graph

ii) An upper estimate for the area under the given graph

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