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The Trapezoidal Rule for Numerical Integration

Theorem  (Trapezoidal Rule)  Consider over , where . The trapezoidal rule is

.

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

.

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

.

Proof  Trapezoidal Rule for Numerical Integration  Trapezoidal Rule for Numerical Integration

Composite Trapezoidal Rule

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

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

.

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

.

Remainder term for the Composite Trapezoidal Rule

Corollary  (Trapezoidal Rule: Remainder term)  Suppose that is subdivided into  m  subintervals    of width  .   The composite trapezoidal 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 (Trapezoidal Rule  Trapezoidal Rule).

Algorithm Composite Trapezoidal Rule.  To approximate the integral

,

by sampling at the equally spaced points    for  ,  where  .  Notice that    and  .

Mathematica Subroutine (Trapezoidal Rule).

Or you can use the traditional program.

Mathematica Subroutine (Trapezoidal Rule).

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

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

Example 7.  Numerically approximate the integral by using the trapezoidal 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 trapezoidal 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.

Recursive Integration Rules

Theorem (Successive Trapezoidal Rules)  Suppose that    and the points    subdivide into    subintervals equal width  .  The trapezoidal rules obey the relationship

.

Definition (Sequence of Trapezoidal Rules)  Define  ,  which is the trapezoidal rule with step size  .  Then for each    define    is the trapezoidal rule with step size  .

Corollary (Recursive Trapezoidal Rule)  Start with  .  Then a sequence of trapezoidal rules    is generated by the recursive formula

for  .

where  .

The recursive trapezoidal rule is used for the Romberg integration algorithm.

Various Scenarios and Animations for the Trapezoidal Rule.

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

Animations (Trapezoidal Rule  Trapezoidal Rule).

Trapezoidal Rule for Numerical Integration  Trapezoidal Rule for Numerical Integration  Internet hyperlinks to web sites and a bibliography of articles.

(c) John H. Mathews 2004