Numerical Integration -- the Left and Right Rectangle Rules and the Midpoint Rule

Let be a function defined for all x in the closed interval [a,b]. We may approximate the integral as follows: Divide the integration interval [a,b] into n equal pieces. Let x₀ = a and x₁ = a + (b-a)/n denote the endpoints of the first piece. In similar manner, the endpoints of the jth piece are x_j-1 and x_j , with x_i = a + i*(b-a)/n for i = 0,1,..., n. The width of each piece is = (b-a)/n .

For the left rectangle rule, we approximate the function over the subinterval [x_j-1 , x_j] by its value at the left endpoint of that subinterval, i.e., by . This results in the approximation

Similarly, for the right rectangle rule we approximate the function over the interval [x_j-1 , x_j] by , and for the midpoint rule by , where , resulting in the approximations

and

respectively.

Left Rectangle Rule	Right Rectangle Rule	Midpoint Rule

The formulas look very similar, and require about the same amount of coding and computation. But it can be shown that, for most cases, the midpoint rule formula yields a more accurate approximation than the other two rules (for a given n) and hence is the method of choice among these three.

If your browser is Java-capable, check out this applet which graphically illustrates left and right rectangle rules, and also evaluates the approximations resulting from the application of these rules (click on "Show Riemann sum" in the applet window to activate this feature). Note that you define the function to be plotted and integrated; be sure the variable you use in your definition is x.

Here is another applet which illustrates three numerical integration methods (including the right rectangle rule, which is simply called "Rectangular", and the midpoint rule) for several pre-defined functions.

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