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Trapezoid Rule and Simpson’s Rule Consider the following
Chapter 5, Problem 31E(choose chapter or problem)
Trapezoid Rule and Simpson's Rule Consider the following integrals and the given values of n.
(a) Find the Trapezoid Rule approximations to the integral using n and 2n subintervals.
(b) Find the Simpson's Rule approximation to the integral using 2n subintervals. It is easiest to obtain Simpson's Rule approximations from the Trapezoid Rule approximations, as in Example 6.
(c) Compute the absolute errors in the Trapezoid Rule and Simpson's Rule with 2n subintervals.
\(\int_{0}^{1} e^{2 x} d x\) ; n=25
Questions & Answers
QUESTION:
Trapezoid Rule and Simpson's Rule Consider the following integrals and the given values of n.
(a) Find the Trapezoid Rule approximations to the integral using n and 2n subintervals.
(b) Find the Simpson's Rule approximation to the integral using 2n subintervals. It is easiest to obtain Simpson's Rule approximations from the Trapezoid Rule approximations, as in Example 6.
(c) Compute the absolute errors in the Trapezoid Rule and Simpson's Rule with 2n subintervals.
\(\int_{0}^{1} e^{2 x} d x\) ; n=25
ANSWER:Problem 31E
Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.
a. Find the Trapezoid Rule approximation to the integral using n and 2n subintervals.
b. Find the Simpson’s Rule approximation to the integral using 2n subintervals. It is easiest to obtain Simpson’s Rule approximations from the Trapezoid Rule approximations, as in Example 6.
c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.
Solution
Step 1:
Given integral is
(a)Trapezoidal rule states that
We have a=0, b=, n=25.
Therefore
Divide interval [0, into n=25 subintervals of length with the following endpoints
Now, we just evaluate function at those endpoints:
….
Finally, just sum up above values and multiply by
Answer :