   Chapter 4.2, Problem 5E

Chapter
Section
Textbook Problem

# The graph of a function f is given. Estimate ∫ 0 10 f ( x ) d x using five subintervals with (a) right endpoints, (b) left endpoints, and (c) midpoints. To determine

Part (a):

To estimate:

010fxdx  using five subintervals with right end points.

Explanation

1) Concept:

Use theorem (4) to estimate the value of the integral by using the right end points.

2) Theorem (4):

If f is integrable on [a, b] then

abf(x)dx=limni=1nf(xi)x

where  x=b - an,n is the number of the subintervals and xi=a+ix

Then an estimate of abf(x)dx using 5 subintervals and right end points is given by

i=15f(xi)x

Where x=b - a5 and xi are right end points of corresponding intervals

3) Given:

The graph:

4) Calculation:

Here, n=5, a=0 and b=10

So, the width of the subintervals is

x=b-an

x=10-05

x=105=2

By using width  x=2 form the subintervals:

0,2, 2, 4, 4, 6, 6, 8 and [8, 10]

So, the right endpoints are x1=2,

To determine

Part (b):

To estimate:

010fxdx  using five subintervals with the left end points.

To determine

Part (c):

To estimate:

010fxdx  using five subintervals with midpoints

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