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To calculate: The value of integral,
Answer to Problem 1P
Solution:
The value of integral,
Explanation of Solution
Given:
Integral,
Formula used:
(1) Formula for h is given by,
(2) Formula for true error percent is,
(3) Formula for relative percent error is,
Calculation:
Consider the following integral:
This integral is a definite integral.
For analytical solution, simplify it as follows with respect to
Hence, the analytical value of
Let
Use the application of the trapezoidal rule:
For
,
Here,
Formula for h is given by,
Substitute the value of
Values for
Substitute the values in
Calculate
Calculate
Substitute the values
For
,
Formula for h is given by,
Substitute the value of
Values for
,
Substitute the values in
Calculate
Substitute the values
For
,
Substitute the value of
and simplify it as follows,
Values for
Values for
And,
Substitute the values in
Calculate
Calculate
Substitute the values
For
,
Formula for h is given by,
Substitute the value of
Values for
Values for
Values for
Values for
And,
Substitute the values in
Calculate
Calculate
Calculate
Calculate
Substitute the values obtained in equation (4) and simplify it as follows,
Use the Romberg integration method,
For
with
The
represents more accurate value and
represents the less accurate value.
Values for
Substitute the values in
and simplify it as follows,
For
with
Values for
Substitute the values in
For
with
Values for
Substitute the values in
and simplify it as follows,
For
with
Values for
Substitute the values in
and simplify it as follows,
For
with
Values for
Substitute the values in
and simplify it as follows,
For
with
Values for
Substitute the values in
and simplify it as follows,
Formula for true error percent is,
Consider the following values,
Substitute the values and simplify it as follows,
Formula for true error percent for
Consider the following values,
Substitute the values and simplify it as follows,
Formula for true error percent for
Consider the following values,
Substitute the values and simplify it as follows,
Formula for true error percent for
Consider the following values,
Substitute the values and simplify it as follows,
Formula for relative error percent for
Here, the value of
Consider the following values,
Substitute the values and simplify it as follows,
Formula for relative error percent for
Here, the value of
Consider the following values,
Substitute the values and simplify it as follows,
Formula for relative error percent for
Here, the value of
Substitute the value of,
Substitute the values and simplify it as follows,
Consider the following table that shows
Hence, the value of integral,
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Chapter 22 Solutions
Numerical Methods for Engineers
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