# The integral of ∫ 1 2 ( x 3 ) d x

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 5.2, Problem 25E
To determine

## To Find: The integral of ∫12(x3)dx

Expert Solution

The integral of 12(x3)dx=154

### Explanation of Solution

Given information:

The integral is:

12(x3)dx

Concept Used:

If f(x) is integrable on [a,b] , then

abf(x)dx=limni=1nf(xi)Δx

Where Δx=ban and xi=a+iΔx

Calculation:

The integral is obtained as:

12(x3)dxΔx=banΔx=21nΔx=1nxi=a+iΔxxi=1+inf(xi)=(1+in)3

And

Δx=1nlimni=1nf(xi)Δxlimni=1n(1+in)31nlimni=1n(1+i3n3+3in+3i2n2)1nlimn(n+n2(n+1)24n3+3n(n+1)2n+3n(n+1)(2n+1)6n2)1nlimn(n+n2(n+1)24n3+3n(n+1)2n+3n(n+1)(2n+1)6n2)1n=154

Hence,

The integral of 12(x3)dx=154

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