Numerical Integration with the Trapezoidal Rule

    The trapezoidal rule uses trapezoids instead of rectangles to approximate the definite interval over a closed bounded interval. By using points on the graph of the function determined by a uniform width partition of the interval the upper boundary of the trapezoid is formed. Recall that the area of a trapezoid is one half the width times the sum of the two heights; and by using this formula we can add up the areas to approximate the definite integral. Of course the more subintervals, (or said another way: the more trapezoids) the more accuracy of the estimation.

    Consider on and say we want to find the area. We could use rectangles as in a Riemann sum; however from inspection of the following diagram it might seem more reasonable to use trapezoids:



The area is underestimated but not as much as using rectangles would (with the same partition).

Definition (Trapezoidal Rule) The trapezoidal rule estimates the definite integral of over using the formula,



where and

The trapezoidal rule formula can be simplied as follows,









which is particularly nice if you want to program or pass a limit.

Example (Trapezoidal Rule) Consider on and let   be a partition of the interval (so ).





which is a much better estimation than using rectangles with the same partition.

Definition (Error in the Trapezoidal Rule) If has a continuous second derivative on then the error in approximating   by the trapezoidal rule satisfies:



where is the maximum value of on

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Cite this as:
Numerical Integration With The Trapezoidal Rule
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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