   Chapter 4.3, Problem 5E

Chapter
Section
Textbook Problem

Evaluating a Definite Integral as a Limit In Exercises 5-10, evaluate the definite integral by the limit definition. ∫ 2 6 8   d x

To determine

To calculate: Definite integral 268dx by the limit definition.

Explanation

Given: 268dx

Formula used:

Definite integral of f(x) from a to b:abf(x)dx=limΔ0i=1nf(ci)Δxi

where 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

where c is a constant.

Calculation:

The function f(x)=8 can be integrated on the interval [2,6] because it is continuous on [2,6]. Definition of integrability means that any partition whose norm approaches 0 can be used to determine the limit. For computational convenience, define Δ by subdividing [2,6] into n subintervals of equal width as below:

Δxi=62n=4n

Choose ci as the right endpoints of each subintervals produces

ci=2+i(Δx)=2+i(4n)=4i+2nn

The definite integral is:

268dx=limΔ0i=1nf(ci)Δxi

Since, ci=(4i+2nn)

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