   Chapter 4.3, Problem 10E

Chapter
Section
Textbook Problem

Evaluating a Definite Integral as a Limit In Exercises 5-10, evaluate the definite integral by the limit definition. ∫ − 2 1 ( 2 x 2 + 3 )   d x

To determine

To calculate: Definite integral of 21(2x2+3)dx by the limit definition.

Explanation

Given: 21(2x2+3)dx

Formula used:

Formula for the 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.

Formula for the sum of a constant n times:i=1nc=nc where c is a constant.

Formula for the sum of first n terms: i=1ni=n(n+1)2

Formula for the sum of squares of first n terms:i=1ni2=n(n+1)(2n+1)6

Calculation: Function f(x)=2x2+3 can be integrated on the interval [2,1] because it is continuous on [2,1].

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

For computational convenience, define Δ by subdividing [2,1] into n subintervals of equal width as below:

Δxi=1(2)n=3n

Choosing ci as the right endpoints of each subintervals produces,

ci=2+i(Δx)=2+i(3n)=3i2nn

So, the definite integral is:

21(2x2+3)dx=limΔ0i=1nf(ci)Δxi

Since, ci=(3i2nn)

limΔ0i=1nf(ci)Δxi=limΔ0i=1nf(3i2nn)Δxi

Use the values of f(3i2nn) and Δxi.

So, the expression becomes,

limΔ0i=1nf(3i2nn)Δxi=limni=1n(2(3i2nn)2+3)(3n)

Factor out 3n3 from the sum

limni=1n(2(3i2nn)2+3)(3n)=limn

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