Improper integrals by numerical methods Use the Trapezoid

Question

Problem 51E

Improper integrals by numerical methods Use the Trapezoid Rule (Section 7.6) to approximate with R =2, 4, and 8. For each value of R, take n = 4, 8, 16, and 32, and compare approximations with successive values of n. Use these approximations to approximate .

Answer 1

Step-by-Step Solution:

Step 1 of 3

Solution:-

Step1

Given that

Use the Trapezoid Rule (Section 7.6) to approximate with R =2, 4, and 8. For each value of R, take n = 4, 8, 16, and 32, . Use these approximations to approximate .

Step2

To find

compare approximations with successive values of n.

Step3

N=4 and R=2

Using trapezoidal rule

$\triangle{}x=\frac{b-a}{n}=\frac{2-0}{4}=\frac{2}{4}=$ $\frac{1}{2}$

Divide interval [0,2] into n=4 subintervals of length $\triangle{}x=\frac{1}{2}$

a=0, $\frac{1}{2},1,\frac{3}{2},2=b$

f( ${x}_{0}^{\ })=f(a)=f(0)=1=1$

2f( ${x}_{1}^{\ })=2f(\frac{1}{2})=\frac{2}{{e}_{\ }^{\frac{1}{2}}}=1.5576$

2f( ${x}_{2}^{\ })=2f(1)=\frac{2}{e}=0.7357$

2f( ${x}_{3}^{\ })=2f(\frac{3}{2})=\frac{2}{{e}_{\ }^{\frac{9}{4}}}=0.2108$

f( ${x}_{4}^{\ })=f(b)=f(2)={e}_{\ }^{-4}=0.0183$

Finally , just sum up above values and multiply by $\frac{\Delta{}x}{2}=\frac{1}{4}$

= $\frac{1}{4}(1+1.5576+0.7357+0.2108+0.01832)$

= $\frac{3.52242}{4}=0.880605$

Step4

N=8 and R=4

Using trapezoidal rule

$\triangle{}x=\frac{b-a}{n}=\frac{4-0}{8}=\frac{4}{8}=\frac{1}{2}$

Divide interval [0,4] into n=8 subintervals of length $\triangle{}x=\frac{1}{2}$

a=0, $\frac{1}{2},1,\frac{3}{2},2,\frac{5}{2},3,\frac{7}{2},4=b$

f( ${x}_{0}^{\ })=f(a)=f(0)=1=1$

2f( ${x}_{1}^{\ })=2f(\frac{1}{2})=1.5576$

2f( ${x}_{2}^{\ })=2f(1)=\frac{2}{e}=0.7357$

2f( ${x}_{3}^{\ })=2f(\frac{3}{2})=\frac{2}{{e}_{\ }^{\frac{9}{4}}}=0.2108$

2f( ${x}_{4}^{\ })=2f(2)=0.0366$

2f( ${x}_{5}^{\ })=2f(\frac{5}{2})=0.0039$

2f( ${x}_{6}^{\ })=2f(3)=0.00025$

2f( ${x}_{7}^{\ })=2f(\frac{7}{2})=9.570.{10}_{\ }^{-6}$

f( ${x}_{8}^{\ })=f(b)=f(4)=1.125.{10}_{\ }^{-7}$

Finally , just sum up above values and multiply by $\frac{\Delta{}x}{2}=\frac{1}{4}$

= $\frac{1}{4}(1+1.5576+0.7357+0.2108+0.0366+0.0039+0.00025+9.570.{10}_{\ }^{-6}+1.125.{10}_{\ }^{-7})$

=0.886227

Step5

N=16 and R=8

Using trapezoidal rule

$\triangle{}x=\frac{b-a}{n}=\frac{8-0}{16}=\frac{8}{16}=\frac{1}{2}$

Divide interval [0,8] into n=16 subintervals of length $\triangle{}x=\frac{1}{2}$

a=0, $\frac{1}{2},1,\frac{3}{2},2,\frac{5}{2},3,\frac{7}{2},4.....\frac{15}{2},8=b$

f( ${x}_{0}^{\ })=f(a)=f(0)=1=1$

2f( ${x}_{1}^{\ })=2f(\frac{1}{2})=1.5576$

2f( ${x}_{2}^{\ })=2f(1)=\frac{2}{e}=0.7357$

2f( ${x}_{3}^{\ })=2f(\frac{3}{2})=\frac{2}{{e}_{\ }^{\frac{9}{4}}}=0.2108$

.

2f( ${x}_{15}^{\ })=2f(\frac{15}{2})=7.4467.{10}_{\ }^{-25}$

f( ${x}_{16}^{\ })=f(b)=f(8)=1.6038.{10}_{\ }^{-28}$

Finally , just sum up above values and multiply by $\frac{\Delta{}x}{2}=\frac{1}{4}$

= $\frac{1}{4}(1+1.5576+0.7357+...............+7.4467.{10}_{\ }^{-25}+1.6038.{10}_{\ }^{-28})$

=0.886227

Step 2 of 3

Chapter 7.7, Problem 51E is Solved

Step 3 of 3

Improper integrals by numerical methods Use the Trapezoid

Solution for problem 51E Chapter 7.7

Chapter 7.7, Problem 51E is Solved

Textbook: Calculus: Early Transcendentals

Edition: 1

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

ISBN: 9780321570567

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