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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Using the Trapezoidal Rule and Simpson’s Rule In Exercises 1-10, use the Trapezoidal Rule and Simpson’s Rule to approximate the value of the definite integral for the indicated value of n. Compare these results with the exact value of the definite integral. Round your answers to four decimal places. See Examples 1 and 2.

0 2 e 4 x d x , n = 8

To determine

To calculate: The value of the integral 02e4xdx,n=8 by use of the Trapezoidal Rule and Simpson’s Rule to approximate the value of the definite integral for the indicated value of n. Compare these results with the exact value of definite integral. Round your answers to four decimal places.

Explanation

Given Information:

The definite integral is 02e4xdx,n=8.

Formula used:

1. Trapezoidal Rule:

If a function f is continuous on [a,b], then

abf(x)dx(ba2n)[f(x0)+2f(x0)+...+2f(xn1)+f(xn)]

2. Simpson’s Rule:

If f is continuous on [a,b] and n is an even integer, then

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

Calculation:

Calculation to get exact value:

Consider the definite integral 02e4xdx,n=8.

02e4xdx=14[e4x]02=14e8+14

Simplify as:

02e4xdx=e8+140.2499

Therefore, the exact value: 02e4xdx0.2499.

Calculation by Trapezoidal Rule:

Consider the definite integral 02e4xdx,n=8.

When n=8, the width of each subinterval is,

208=14

And the end points of subintervals are,

For x0,

x0=0

For x1,

x1=14

For x2,

x2=14+14=12

For x3,

x3=12+14=34

For x4,

x4=34+14=44=1

For x5,

x5=1+14=54

For x6,

x6=54+14=64=32

For x7,

x7=32+14=74

For x8,

x8=74+14=2

By Trapezoidal Rule,

02e4xdx=12×8[e0+2e4×14+2e4×12+2e4×34

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