   Chapter 4.3, Problem 9E

Chapter
Section
Textbook Problem

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

To determine

To calculate: Definite integral of 12(x2+1)dx by the limit definition.

Explanation

Given: 12(x2+1)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)=x2+2 can beintergrated on the interval [1,2] because it is continuous on [1,2].

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 [1,2] into n subintervals of equal width as below:

Δxi=2(1)n=1n

Choosing ci as the right endpoints of each subintervals produces,

ci=1+i(Δx)=1+i(1n)=i+nn

So, the definite integral is:12(x2+1)dx=limΔ0i=1nf(ci)Δxi

Since, ci=(i+nn)

limΔ0i=1nf(ci)Δxi=limΔ0i=1nf(i+nn)Δxi

Use the values of f(i+nn) and Δxi.

So, the expression becomes,

=limΔ0i=1nf(i+nn)Δxi=limni=1n((i+nn)2+1)(1n)

Factor out 1n3 from the sum as below:

limni=1n((i+nn)2

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