   Chapter 4.3, Problem 6E

Chapter
Section
Textbook Problem

Evaluating a Definite Integral as a Limit In Exercises 5-10, evaluate the definite integral by the limit definition. ∫ − 2 3 x   d x

To determine

To calculate: Definite integral of 23xdx by the limit definition.

Explanation

Given: 23xdx

Formula used: Formula for the definite integral of f(x) from a to b:

abf(x)dx=limΔ0i=1nf(ci)Δxi

Here a is the lower limit of integration and b is the upper limit of integration.

The sum of a constant n times is written as:

i=1nc=nc

Here c is a constant.

Calculation: The function f(x)=x can be integrated on the interval [2,3] because it is continuous on [2,3].

Definition of integrability implies that any partition whose norm approaches 0 can be used to determine the limit.

For computational convenience, define Δ by subdividing [2,3] into n subintervals of equal width as:

Δxi=3(2)n=5n

Choose ci as the right endpoints of each subinterval. Therefore,

ci=2+i(Δx)=2+i(5n)=5i2nn

So, the definite integral is:

23xdx=limΔ0i=1nf(ci)Δxi

Since, ci=(5i2nn)

23xdx=limΔ0i=1nf(5i2nn)Δxi

Use value of f(5i2nn) and Δxi

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