# 6. Simpson's Rule

by M. Bourne

The Trapezoidal Rule is an improvement over using rectangles
because we have much less "missing" from our calculations. We
used **straight lines** to model the curve in
trapezoidal Rule.

We seek an even better approximation. In **Simpson's
Rule**, we use **parabolas** to approximate each part of the curve. This
proves to be very efficient. (See more about Parabolas.)

We can show (by integrating the area under each parabola and
adding these areas) that the approximate area is given the following. This is called **Simpson's Rule**:

`"Area"=int_a^bf(x)dx`

`~~(Deltax)/3(y_0+4y_1+2y_2+4y_3+2y_4...+4y_(n-1)+y_n)`

**Note:** In Simpson's Rule, *n* must be EVEN.

**Example**

Approximate** **`int_2^3(dx)/(x+1`using Simpson's Rule with `n=4`.

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