Subject: Calculus

# Simpsons Rule

We go to the third method in numerical integration, the **Simpson’s Rule**. Here, instead of actually using line segments in getting an approximate of a curve, we now use parabolas. We first divide an interval \left [a, b\right ] into n subintervals of equal length. The figure below shows the curve that will be approximated.

Here, we represent the length h= \Delta x= \frac{b-a}{n}, but this time we’ll assume that n is an even number. For each consecutive pair of intervals we approximate the curve y=f(x) \geq 0 by a parabola as shown in the above figure.

Now if we let y_i=f(x_i ), then P_i (x_i,y_i ) is the point on the curve lying above x_i. A typical parabola passes through three consecutive points, P_i, P_{i+1}, and P_{i+2}. For us to be able to solve the problem easier, let us consider the case where x_0= -h, x_1=0 ,and x_2=h. We can illustrate this as follows,

From our elementary algebra, we know that the a parabola through the points P_0, P_1, and P_2 take the form of y=ax^2+ bx+c. We can therefore calculate the area under the parabola by integrating the equation from x = -h to x = h which is

But since the parabola passes through P_0 (h, y_0 ), P_1 (0,y_1 ), and P_2 (h, y_2), we have

and therefore

Thus, the under the parabola can be written as

Now, by shifting this parabola horizontally we do not change the area under it. This means that the area under the parabola through P_0, P_1, and P_2 from x=x_0 to x=x_2in the first figure is still

Similarly, the area under the parabola through P_2, P_3, and 4 from x=x_2 to x=x_4 is

If we compute the areas under all the parabolas in this manner and add the results, we get

Although we have derived this approximation for the case in which x \geq 0 , it is a reasonable approximation for any continuous function f and is called **Simpson’s Rule** after the English mathematician Thomas Simpson (1710–1761). Note the pattern of coefficients: 1, 4, 2, 4, 2, 4, 2, . . . , 4, 2, 4, 1.

So in general, we can summarize the Simpson’s Rule as

where n is even and \Delta x=\frac{b-a}{n}.

## Error Bound for Simpson’s Rule

Suppose that \left \|f^4 (x) \right \| \leq K for a \leq x \leq b. If E_s is the error involved in using Simpson’s Rule, then

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