Concept explainers
(a)
Approximate the value of integral using Simpson’s Rule with
(a)
Answer to Problem 19E
With
The approximate value is 12.
Explanation of Solution
Given information:
Simpson’s rule formula:
Where,
Now,
For
We have
With
Since
Then
For
For
For
For
For
Substitute the above values:
(b)
Exact value of integral and error
(b)
Answer to Problem 19E
Exact value of integral is also 12.
Error,
Explanation of Solution
Given information:
Evaluate the integral:
Thus,
The actual value of integral is 12.
From Part (a),
The estimated value was 12.
Then
Error,
(c)
Prediction of error
(c)
Answer to Problem 19E
From error bound formula,
Explanation of Solution
Given information:
According to error − bound formula:
Since
Then
First derivative:
Second derivative:
Third derivative:
Fourth derivative:
Hence,
Therefore,
(d)
Whether it is possible to make general statement about using Simpson’s Rule to approximate integrals of cubic polynomials.
(d)
Answer to Problem 19E
Simpson rule will always give exact values for all cubic polynomials.
Explanation of Solution
Given information:
For any cubic polynomial,
If
Then
Then
The error − bound formula would give
Such that
Error is always zero.
Therefore,
Simpson rule will always give exact values for all cubic polynomials.
Chapter 6 Solutions
Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)
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