Question 4.7.2: Using a Program for the Midpoint Rule Repeat example 7.1 usi...

Calculus: Early Transcendental Functions

Using a Program for the Midpoint Rule

Repeat example 7.1 using a program to compute the Midpoint Rule approximations for n = 8, 16, 32, 64 and 128.

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You should confirm the values in the following table. We include a column displaying the error in the approximation for each n (i.e., the difference between the exact value of 1 and the approximate values).

n	Midpoint Rule	Error
4	0.984375	0.015625
8	0.99609375	0.00390625
16	0.99902344	0.00097656
32	0.99975586	0.00024414
64	0.99993896	0.00006104
128	0.99998474	0.00001526

You should note that each time the number of steps is doubled, the error is reduced approximately by a factor of 4. Although this precise reduction in error will not occur with all integrals, this rate of improvement in the accuracy of the approximation is typical of the Midpoint Rule.

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