# The integral of ∫ 0 2 ( 2 − x 2 ) 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 23E
To determine

## To Find: The integral of ∫02(2−x2)dx

Expert Solution

The integral of 02(2x2)dx=43

### Explanation of Solution

Given information:

02(2x2)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:

02(2x2)dx

And,

Δx=banΔx=20nΔx=2nxi=a+iΔxxi=2inf(xi)=2(2in)2

Therefore,

Δx=2n

So,

limni=1nf(xi)Δxlimn0n(2(2in)2)2nlimn(48n3×n(n+1)(2n+1)6)limn(48n3×n3(1+1n)(2+1n)6)=43

Hence,

The integral of 02(2x2)dx=43

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