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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 8 x 3   d x ,   n = 8

To determine

The value of the integral 08x3dx,n=8 by using 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 08x3dx,n=8.

Formula used:

1. Trapezoidal Rule:

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

02f(x)dx,n=(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

02f(x)dx,n=(ba3n)[f(x0)+4f(x1)+2f(x2)+4f(x3)++4f(xn1)+f(xn)].

Calculation:

Calculation to get exact value:

Consider the definite integral 08x3dx,n=8.

08x3dx=08x3+223+22        =082x525        =25×852      =252×52

Further simplify:

2535.3333

Calculation by Trapezoidal Rule:

Consider the definite integral 08x3dx,n=8.

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

408=48=12

And the end points of subintervals are,

x0=0,x1=12x2=1

And for x3,

x3=1+12=32

And for x4,

x4=32+12=42=2

And for x5,

x5=2+12=52

And for x6,<

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