Midpoint and Trapezoidal Rules

There are some cases when it is impossible to find the definite integral

They can be approximated by using types of Riemann sums

Approximating Integrals

Midpoints of drawn rectangles are used

where

Midpoint Rule

1 2( ) ( ) ( )... ( )b

naf x dx x f x f x f x

b axn

Uses trapezoids instead of rectangles

Trapezoidal Rule

0 1 2( ) ( ) 2 ( ) 2 ( ) ..... ( )2

b

na

xf x dx f x f x f x f x

Use both the Midpoint and Trapezoidal rules to evaluate the integral with n = 5

Example

2

1

1 dxx