# 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(x_{0 }) + 2 f(x_{1
})
+ 2 f(x_{2 }) + ... + 2 f(x_{n-1 }) + f(x_{n
})
]

* 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) = Ax*^{2 }+ Bx + C be the equation
of
one parabola.

Then

Proof:

Repeating this gives: