Trapezoid rule - approximating definite integrals, Mathematics

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Trapezoid Rule - Approximating Definite Integrals

For this rule we will do similar set up as for the Midpoint Rule. We will break up the interval [a, b] into n subintervals of width,

Δx = b - a / n

After that on each subinterval we will estimated the function with a straight line that is equivalent to the function values at either endpoint of the interval.  Now here is a sketch of this case for n = 6.

2254_Trapezoid Rule - Approximating Definite Integrals.png

Every object in this is a trapezoid (therefore the rule's name...) and like we can see some of them do a very good job of approximating the actual area within the curve and others don't do such a good job.

The area of the trapezoid in the interval [xi-1, xi] is given by,  

Ai = Δx/2 (f (xi-1) + f (xi))

So, if we employ n subintervals the integral is just about,

ba f(x) dx ≈ Δx/2 (f (x0) + f (x1)) + Δx/2 (f(x1) + f (x2)) +.....+ Δx/2 (f (xn-1) + f (xn))

On doing a little simplification we reach at the general Trapezoid Rule.

 ∫ba f (x) dx ≈ Δx/2 [f (x0) + 2f (x1) + 2f (x2) + .... + 2f (xn-1) + f(xn)]

Note: all the function evaluations along with the exception of the first and last, are multiplied by 2.


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