Concept explainers
Evaluate the following
(a) Analytically;
(b) Single application of the trapezoidal rule;
(c) Multiple-application trapezoidal rule, with
(d) Single application of Simpson's 1/3 rule;
(e) Multiple-applicationSimpson's 1/3 rule, with
(f) Single application of Simpson's3/8 rule; and
(g) Multiple-application Simpson's rule, with
For each of the numerical estimates (b) through (g), determine the percent relative error based on (a).
(a)
To calculate: The value of the integral
Answer to Problem 1P
Solution:
The value of integral
Explanation of Solution
Given:
The integral,
Formula used:
Here,
Calculation:
Consider the integral,
The value of the integral is,
Therefore, the value of the integral is
(b)
To calculate: The value of the integral
Answer to Problem 1P
Solution:
The value of integral
Explanation of Solution
Given:
The integral,
The exact value of the integral
Formula used:
Single application version of Trapezoidal rule: If
Percentage error is,
Calculation:
Consider the integral,
Single application version of Trapezoidal rule is,
And
The value of the integral is,
And Percentage error is,
Therefore, the value of the integral is
(c)
To calculate: The value of the integral
Answer to Problem 1P
Solution:
Explanation of Solution
Given:
The integral,
The exact value of the integral
Formula used:
Multiple application version of Trapezoidal rule: If
Here,
Percentage error is,
Calculation:
Consider the integral,
Here, the function is,
Multiple application version of Trapezoidal rule is,
When
Here,
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
Thus, the value of the integral is,
Percentage error is,
Multiple application version of Trapezoidal rule is,
When
Here,
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
Thus, the value of the integral is,
Percentage error is,
Therefore, the value of the integral when
(d)
To calculate: The value of the integral
Answer to Problem 1P
Solution:
Explanation of Solution
Given:
The integral,
The exact value of the integral
Formula used:
Single application version of Simpson’s
Here,
Percentage error is,
Calculation:
Consider the integral,
Here, the function is,
Single application version of Simpson’s
Here,
Here,
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
Thus, the value of the integral is,
Percentage error is,
Therefore, the value of the integral when
(e)
To calculate: The value of the integral
Answer to Problem 1P
Solution:
Explanation of Solution
Given:
The integral,
The exact value of the integral
Formula used:
Multiple application version of Simpson’s
Here,
Percentage error is,
Calculation:
Consider the integral,
Here, the function is,
Multiple application version of Simpson’s
When
Here,
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
Thus, the value of the integral is,
Percentage error is,
Therefore, the value of the integral when
(f)
To calculate: The value of the integral
Answer to Problem 1P
Solution:
Explanation of Solution
Given:
The integral,
The exact value of the integral
Formula used:
Single application version of Simpson’s
Here,
Percentage error is,
Calculation:
Consider the integral,
Here, the function is,
Single application version of Simpson’s
Here,
Here,
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
Thus, the value of the integral is,
Percentage error is,
Therefore, the value of the integral when
(g)
To calculate: The value of the integral
Answer to Problem 1P
Solution:
Explanation of Solution
Given:
The integral,
The exact value of the integral
Formula used:
Multiple application version of Simpson’s
Single application version of Simpson’s
Single application of Simpson’s
Here,
Percentage error is,
Calculation:
Consider the integral,
Here, the function is,
Multiple application version of Simpson’s
Here,
Here,
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
The value of
The value of the function at
Thus, the value of the integral is,
Percentage error is,
Therefore, the value of the integral when
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