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Comparing the Midpoint and Trapezoid Rules pply the
Chapter 5, Problem 23E(choose chapter or problem)
Comparing the Midpoint and Trapezoid Rules Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 7.4 showing the approximations and errors for n=4, 8, 16, and 32. The exact values of the integrals are given for computing the error.
\(\int_{0}^{\pi / 4} 3 \sin 2 x d x=\frac{3}{2}\)
Questions & Answers
QUESTION:
Comparing the Midpoint and Trapezoid Rules Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 7.4 showing the approximations and errors for n=4, 8, 16, and 32. The exact values of the integrals are given for computing the error.
\(\int_{0}^{\pi / 4} 3 \sin 2 x d x=\frac{3}{2}\)
ANSWER:Solution:-
Step1
Given that
Apply the Midpoint and Trapezoid Rules to the following integrals.
n = 4,8, 16, and 32
The exact values of the integrals are given for computing the error.
Step2
To find
Make a table showing the approximations and errors for n = 4,8, 16, and 32.
Step3
a=0 , b=0.7854, (N=4,8,16,32)
Using midpoint rule
Step4
Midpoint sum for N=4
+
+-------+
)
+
+-------+
)
=0.93193
Error for N=4 for midpoint rule
=
=
=0.37871
Step5
Using midpoint rule
Midpoint sum for N=8
+
+-------+
)
+
+-------+
)
=1.2093
Error for N=8 for midpoint rule
=
=
=0.19379
Step6
Using midpoint rule