   Chapter 8.6, Problem 14E ### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

#### Solutions

Chapter
Section ### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Using the Trapezoidal Rule and Simpson's Rule In Exercises 3-14, use the Trapezoidal Rule and Simpson’s Rule to approximate the value of the definite integral for the given value of n. Round your answer to four decimal places and compare the results with the exact value of the definite integral. ∫ 0 2 x ln ( x + 1 ) d x ,   n = 4

To determine

To calculate: The approximate value of definite integral 02xln(x+1)dx for the specified value of n=4 by Trapezoidal Rule and Simpson’s Rule and compare it with the exact value of the definite integral.

Explanation

The integral is 02xln(x+1)dx.

Formula used:

Trapezoidal Rule is:

abf(x)dxba2n[f(x0)+2f(x1)+....+2f(xn1)+f(xn)]

Simpson’s Rule is:

abp(x)dx(ba3n)[f(x0)+4f(x1)+2f(x2)+4f(x3)+...2f(xn2)+4f(xn1)+f(xn)]

Calculation:

Suppose f(x)=xln(x+1).

Therefore, add 204=12 to each term to get values of x0,x2.......xm where x0=0 and x4=2.

Thus, the values of x are:

x1=0+12=12x2=12+12=1

And,

x3=1+12=32x4=32+12=2

Now, by trapezoidal Rule:

Here b=2, a=0 and n=4. Therefore,

02xln(x+1)dx202(4)[f(0)+2f(12)+2f(1)+2f(32)+f(2)]=14[((0)ln(0+1))+2((12)ln(12+1))+2(1ln(1+1))+2((32)ln(32+1))+(2ln(2+1))]=14[0+ln32+2ln2+3ln52+2ln3]=14[0.4054+1.38629+2.7489+2.19722]

By simplifying more,

02xln(x+1)dx=6.73781441.6845

Hence, the value of integral by Trapezoidal Rule is 1.6845.

Now, by Simpson’s Rule:

Here, b=2, a=0 and n=4.

Therefore, the integral becomes,

02xln(x+1)dx203(4)[f(0)+4f(12)+2f(1)+4f(32)+f(2)]=16[((0)ln(0+1))+4((12)ln(12+1))+2(1ln(1+1))+4((32)ln(32+1))+(2ln(2+1))]=16[0+2ln32+2ln2+6ln52+2ln3]=16[0.8108+1.38629+5.4977+2.19722]

By simplifying more,

02xln(x+1)dx=16[9.8916]1

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

## Additional Math Solutions

#### Find the mean for the following set of scores: 2, 7, 9, 4, 5, 3, 0, 6

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

#### 21. For the function graphed in Exercise 18, for what values of x does ?

Mathematical Applications for the Management, Life, and Social Sciences

#### Simplify: 1518

Elementary Technical Mathematics 