Trapezoidal Rule and Simpson's Rule

Trapezoidal Rule

Divide the area under the curve into n trapezoids from x = a to x = b.  The area can be found with the following formula:

T = b - a   [ f(x0 ) + 2 f(x1 ) + 2  f(x2 ) + ... + 2 f(xn-1 ) + f(xn ) ]
        2n
The proof comes from using the formula for the area of a trapezoid over and over.  Also, h = b - a.
                                                                                                                                             n


 

Simpson's Rule

Divide the area under the curve into n (where n is even) parabolas from x = a to x = b.  The area can be found with the following formula:

Let P(x) = Ax2 + Bx + C be the equation of one parabola.
Then 

Proof:

  Repeating this gives: