   Chapter 13.8, Problem 8E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# For each integral in Problems 7-12, do the following.(a) Approximate its value using the Trapezoidal Rule.(b) Approximate its value using Simpson’s Rule.(c) Find its exact value by integration.(d) State which approximation is more accurate. (Round each result to 2 decimal places.) ∫ 1 0   x 3 d x ;    n = 4

(a)

To determine

To calculate: The approximate value of integral 01x3dx for n=4 by Trapezoidal Rule.

Explanation

Given information:

The provided integral 01x3dx for n=4.

Formula used:

The length of each n subdivision for a closed interval [a,b].

h=ban

The approximate value of integral abf(x)dx for n equal subdivision of closed interval [a,b] by Trapezoidal Rule is,

abf(x)dxh2[f(x0)+2f(x1)+2f(x2)+2f(x3)++f(xn)]

Where h is the length of each subdivision.

Calculation:

Consider the provided integral 01x3dx for n=4.

Since the number of equal subdivision is 4 and the lower and upper limit of the integral are 0 and 1 respectively.

n=4a=0b=1

Use the formula for the length of each n subdivision for a closed interval [a,b].

h=ban

Substitute 0 for a, 1 for b and 4 for n.

h=104h=14

The corresponding value of xi=a+ih and f(xi)=xi2 for a=0 and h=14 is shown below

(b)

To determine

To calculate: The approximate value of integral 01x3dx for n=4 by Simpson Rule.

(c)

To determine

To calculate: The exact value of 01x3dx by integration.

(d)

To determine

The most accurate approximation rule among the Trapezoidal rule and Simpson Rule.

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