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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 2 1 x 2 + 1 d x , n = 8

(a)

To determine

The value of the integral 021x2+1dx,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 four decimal places.

Explanation

Given Information:

The definite integral is 021x2+1dx,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:

Consider the definite integral,

021x2+1dx,n=8

Calculation by Trapezoidal Rule:

When, n=8 the width of each subinterval is

208=28=14

And the end points of subintervals are,

x0=0x1=0+14=14

And,

x2=14+14=12x3=12+14=34

And,

x4=34+14=1x5=1+14=54

And,

x6=54+14=32x7=32+14=74

And,

x8=74+14

(b)

To determine

The value of the integral 021x2+1dx,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 four decimal places.

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