Chapter 8.6, Problem 12E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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 x 2 + 1 d x ,   n = 4

To determine

To calculate: The approximate value of definite integral 02xx2+1dx 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

Given:

The integral is âˆ«02xx2+1dx.

Formula used:

Trapezoidal Rule is:

âˆ«abf(x)dxâ‰ˆbâˆ’a2n[f(x0)+2f(x1)+....+2f(xnâˆ’1)+f(xn)]

Simpsonâ€™s Rule is:

âˆ«abp(x)dxâ‰ˆ(bâˆ’a3n)[f(x0)+4f(x1)+2f(x2)+4f(x3)+...2f(xnâˆ’2)+4f(xnâˆ’1)+f(xn)]

Calculation:

Suppose f(x)=xx2+1

Since n=4

Therefore, add 2âˆ’04=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,

âˆ«02xx2+1dxâ‰ˆ2âˆ’02(4)[f(0)+2f(12)+2f(1)+2f(32)+f(2)]=14[(002+1)+2((12)(1/2)2+1)+2(112+1)+2((32)(3/2)2+1)+(222+1)]=14[0+2((12)54)+2(2)+2((32)134)+(25)]=14[0+52+22+3213+25]

By simplifying more,

âˆ«02xx2+1dx=14[1

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