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Get Full Access to Calculus: Early Transcendentals - 1 Edition - Chapter 7.6 - Problem 23e
Get Full Access to Calculus: Early Transcendentals - 1 Edition - Chapter 7.6 - Problem 23e

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# Comparing the Midpoint and Trapezoid Rules pply the ISBN: 9780321570567 2

## Solution for problem 23E Chapter 7.6

Calculus: Early Transcendentals | 1st Edition

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Problem 23E

Problem 23E

Comparing the Midpoint and Trapezoid Rules

pply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 7.4 showing the approximations and errors for n = 4,8, 16, and 32. The exact values of the integrals are given for computing the error. Step-by-Step Solution:
Step 1 of 3

Solution:-

Step1

Given that

Apply the Midpoint and Trapezoid Rules to the following integrals.

n = 4,8, 16, and 32

The exact values of the integrals are given for computing the error. Step2

To find

Make a table  showing the approximations and errors for n = 4,8, 16, and 32.

Step3

a=0 , b= 0.7854, (N=4,8,16,32)

Using midpoint rule Step4

Midpoint sum for N=4  + +-------+ )  + +-------+ ) =0.93193

Error for N=4 for midpoint rule = = =0.37871

Step5

Using midpoint rule Midpoint sum for N=8  + +-------+ )  + +-------+ ) =1.2093

Error for N=8 for midpoint rule = = =0.19379

Step6

Using midpoint rule  Midpoint sum for N=16  + +-------+ )  + +-------+ ) =1.3536

Error for N=16 for midpoint rule = = =0.0976

Step7

Using midpoint rule 0.024544

Midpoint sum for N=32  + +-------+ )  + +-------+ ) =1.4266

Error for N=32 for midpoint rule = =0.048918

Step8

Table for midpoint approximation and error

 N Approximation Error 4 0.93193 0.37871 8 1.2093 0.19379 16 1.3536 0.0976 32 1.4266 0.048918

Step9

N=4

Using trapezoidal rule 0.19635

Trapezoidal  sum for N=4   =1.4952

Error for N=4 for Trapezoidal  rule = =0.012884204

Step10

N=8

Using trapezoidal rule  Trapezoidal  sum for N=8   =1.4952

Error for N=8 for Trapezoidal  rule = =0.0032148

Step11

N=16

Using trapezoidal rule  Trapezoidal  sum for N=16   =1.4988

Error for N=16 for Trapezoidal  rule = =0.00080332

Step12

N=32

Step 2 of 3

Step 3 of 3

##### ISBN: 9780321570567

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