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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 41–46, approximate the value of the definite integral using (a) the Trapezoidal Rule and (b) Simpson’s Rule for the indicated value of n. Round your answers to three decimal places.

0 8 3 x 2 + 2 d x , n = 8

(a)

To determine

To calculate: The value of the integral 083(x2+2)dx,n=8 by using the Trapezoidal Rule and to approximate the value of the definite integral for the indicated value of n. Round your answers to three decimal places.

Explanation

Given Information:

The definite integral is 083(x2+2)dx,n=8.

Formula used:

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)]

Calculation:

Calculation by Trapezoidal Rule:

Consider the definite integral 083(x2+2)dx,n=8.

When n=8, the width of each subinterval is

808=88=1

And the end points of subintervals are

x0=0,x1=0+1=1,x2=1+1=2,x3=2+1=3,x4=3+1=4,x5=4+1=5,x6=5+1=6,x7=6+1=7,x8=7+1=8

By Trapezoidal Rule

(b)

To determine

To calculate: The value of the integral 083(x2+2)dx,n=8 by using the Simpson’s Rule and to approximate the value of the definite integral for the indicated value of n. Round your answers to three decimal places.

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