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The Midpoint Rule for Numerical Integration

Theorem  (Midpoint Rule)  Consider over , where . The midpoint rule is

.

This is an numerical approximation to the integral of over and we have the expression

.

The remainder term for the midpoint rule is  ,  where lies somewhere between , and have the equality

.
Proof  The Midpoint Rule  The Midpoint Rule

Composite Midpoint Rule

An intuitive method of finding the area under a curve y = f(x)  is by approximating that area with a series of rectangles that lie above the intervals  .  When several rectangles are used, we call it the composite midpoint rule.

Theorem  (Composite Midpoint Rule)  Consider over .  Suppose that the interval is subdivided into  m  subintervals    of equal width    by using the equally spaced nodes    for  .   The composite midpoint rule for m subintervals is

.

This is an numerical approximation to the integral of over and we write

.

Remainder term for the Composite Midpoit Rule

Corollary  (Midpoint Rule: Remainder term)  Suppose that is subdivided into  m  subintervals    of width  .   The composite midpoint rule

is an numerical approximation to the integral, and

.

Furthermore, if ,  then there exists a value  c  with  a < c < b  so that the error term    has the form

.

This is expressed using the "big " notation  .

Remark.  When the step size is reduced by a factor of the error term   should be reduced by approximately .

Animations (Midpoint Rule  Midpoint Rule).

Computer Programs  The Midpoint Rule  The Midpoint Rule

Algorithm Composite Midpoint Rule.  To approximate the integral

,

by sampling at the equally spaced points    for  ,  where  .

Mathematica Subroutine (Midpoint Rule).

Or you can use the traditional program.

Mathematica Subroutine (Midpoint Rule).

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Example 1.  Numerically approximate the integral    by using the midpoint rule with  m = 1, 2, 4, 8, and 16  subintervals.
Solution 1.

Example 2.  Numerically approximate the integral    by using the midpoint rule with  m = 50, 100, 200, 400  and 800  subintervals.
Solution 2.

Example 3.  Find the analytic value of the integral    (i.e. find the "true value").
Solution 3.

Example 4.  Use the "true value" in example 3 and find the error for the midpoint rule approximations in example 2.
Solution 4.

Example 5.  When the step size is reduced by a factor of the error term   should be reduced by approximately  .  Explore this phenomenon.
Solution 5.

Example 6.  Numerically approximate the integral by using the midpoint rule with  m = 1, 2, 4, 8, and 16  subintervals.
Solution 6.

Example 7.  Numerically approximate the integral by using the midpoint rule with  m = 50, 100, 200, 400  and 800  subintervals.
Solution 7.

Example 8.  Find the analytic value of the integral    (i.e. find the "true value").
Solution 8.

Example 9.  Use the "true value" in example 8 and find the error for the midpoint rule approximations in exercise 7.
Solution 9.

Example 10.  When the step size is reduced by a factor of the error term   should be reduced by approximately  .  Explore this phenomenon.
Solution 10.

Various Scenarios and Animations for the Midpoint Rule.

Example 11.  Let    over  .  Use the Midpoint Rule to approximate the value of the integral.
Solution 11.

Animations (Midpoint Rule  Midpoint Rule).

Midpoint Rule  Midpoint Rule  Internet hyperlinks to web sites and a bibliography of articles.

(c) John H. Mathews 2004