Concept explainers
(a)
To calculate: The given integral
(a)
Answer to Problem 9P
Solution: The value of
Explanation of Solution
Given Information:
The given integral is,
Formula used:
Simpson’s 1/3 rule.
Extended Midpoint rule for
Calculation:
Calculate the analytical value of integral,
Rewrite the integral for finite intervals,
The function values are given in table below,
t | 0.0625 | 0.125 | 0.1875 | 0.25 | 0.3125 | 0.375 | 0.4375 | 0.5 |
f(t) | 0.941176 | 0.842105 | 0.761905 | 0.695652 | 0.64 | 0.592593 | 0.551724 | 0.516129 |
Apply numerical integration to simplify,
Substitute the values from above table.
Hence, the value of
(b)
To calculate: The given integral
(b)
Answer to Problem 9P
Solution: The value of
Explanation of Solution
Given Information:
The given integral is,
Formula used:
Simpson’s 1/3 rule.
Extended Midpoint rule for
Calculation:
Calculate the analytical value of integral,
Since
. So integral is,
The function values are given in table below,
x | 0 | 0.0625 | 0.125 | 0.1875 | 0.25 | 0.3125 | 0.375 | 0.4375 | 0.5 |
f(x) | 0 | 0.048 | 0.139 | 0.219 | 0.26 | 0.258 | 0.222 | 0.168 | 0.112 |
Apply 4-application Simpson’s 1/3 rule for first part of integral,
Here,
Substitute the values from above table,
Rearrange the integral for calculation of second integral,
The function values are changed for rearranged integral which is,
t | 0 | 1 | 2 | 3 |
f(t) | 0 | 0.0908 | 0.00142 | 0.303 |
Apply extended midpoint rule with
Substitute the values from above table,
The total integral is,
Substitute the value from above,
Hence, the value of
(c)
To calculate: The given integral
(c)
Answer to Problem 9P
Solution: The value of
Explanation of Solution
Given Information:
The given integral is,
Formula used:
Simpson’s 1/3 rule.
Extended Midpoint rule for
Calculation:
Calculate the analytical value of the integral,
Recall the formula,
Simplify further,
Substitute
Thus, the final value of integral is,
Hence, the value of
The function values are given in table below,
x | 0 | 0.0625 | 0.125 | 0.1875 | 0.25 | 0.3125 | 0.375 | 0.4375 | 0.5 |
f(x) | 1 | 0.9127 | 0.711 | 0.4995 | 0.333 | 0.2191 | 0.1448 | 0.0972 | 0.0667 |
Apply 4-application Simpson’s 1/3 rule for first part of integral,
Here,
Substitute the values from above table,
Rearrange the integral for calculation of second integral,
The function values are changed for rearranged integral which is,
t | 0 | 1 | 2 | 3 |
f(t) | 0.007722 | 0.063462 | 0.148861 | 0.232361 |
Apply extended midpoint rule with
Substitute the values from above table,
The total integral is,
Substitute the value from above,
Hence, the value of
(d)
To calculate: The given integral
(d)
Answer to Problem 9P
Solution: The value of
Explanation of Solution
Given Information:
The given integral is,
Formula used:
Simpson’s 1/3 rule.
Extended Midpoint rule for
Calculation:
Calculate the analytical value of the integral,
Apply Numerical integration to simplify,
Further simplify,
Hence, the value of
The function values are given in table below,
x | 0 | 0.0625 | 0.125 | 0.1875 | 0.25 | 0.3125 | 0.375 | 0.4375 | 0.5 |
f(x) | -14.78 | -6.72 | -2.72 | 0.824 | 0 | 0.303 | 0.368 | 0.335 | 0.2707 |
Apply 4-application Simpson’s 1/3 rule for first part of integral,
Here,
Substitute the values from above table,
Rearrange the integral for calculation of second integral,
The function values are changed for rearranged integral which is,
t | 0 | 1 | 2 | 3 |
f(t) | 0.000461 | 0.073241 | 0.1335696 | 1.214487 |
Apply extended midpoint rule with
Substitute the values from above table,
The total integral is,
Substitute the value from above,
Hence, the value of
(e)
To calculate: The given integral
(e)
Answer to Problem 9P
Solution: The value of
Explanation of Solution
Given Information:
The given integral is,
Formula used:
Simpson’s 1/3 rule.
Extended Midpoint rule for
Calculation:
Calculate the given integral,
Rewrite the given integral,
Recall the formula,
Apply Analytical integration to simplify for
Hence, the value of
The function values are given in table below,
x | 0 | 0.0625 | 0.125 | 0.1875 | 0.25 | 0.3125 | 0.375 | 0.4375 | 0.5 |
f(x) | 0.399 | 0.387 | 0.352 | 0.301 | 0.242 | 0.183 | 0.130 | 0.086 | 0.054 |
Apply 4-application Simpson’s 1/3 rule for first part of integral,
Here,
Substitute the values from above table,
Rearrange the integral for calculation of second integral,
The function values are changed for rearranged integral which is,
t | 0 | 1 | 2 | 3 |
f(t) | 0 | 0 | 0.024413063 | 0.152922154 |
Apply extended midpoint rule with
Substitute the values from above table,
The total integral is,
Substitute the value from above,
Hence, the value of
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Chapter 22 Solutions
Numerical Methods for Engineers
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