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Solution: Comparing the Midpoint and Trapezoid Rules Compare
Chapter 5, Problem 41E(choose chapter or problem)
Comparing the Midpoint and Trapezoid Rules Compare the errors in the Midpoint and Trapezoid Rules with n=4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given).
\(\int_{0}^{\pi / 2} \cos ^{9} x d x=\frac{128}{315}\)
Questions & Answers
QUESTION:
Comparing the Midpoint and Trapezoid Rules Compare the errors in the Midpoint and Trapezoid Rules with n=4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given).
\(\int_{0}^{\pi / 2} \cos ^{9} x d x=\frac{128}{315}\)
ANSWER:Solution:-
Step1
Given that
Compare the errors in the Midpoint and Trapezoid Rules with n =4, 8, 16, and 32 subintervals
n = 4,8, 16, and 32
The exact values of the integrals are given for computing the error.
dx= 0.406349
Step2
To find
Compare the errors in the Midpoint and Trapezoid Rules with n =4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given).
Step3
a=0 , b=1.5708, n=4
Using midpoint rule
Step4
Midpoint sum for N=4
+
+-------+
)
+
+-------+
)
=0.406349
Error for N=4 for midpoint rule
=
=
=1.46785
Step5
Using midpoint rule
Midpoint sum for N=8
+
+-------+
)
+
+-------+
)
=0.406349
Error for N=8 for midpoint rule
=
=
=1.95089
Step6