Area Under a curve5.1

Error

When we use the midpoint rule or trapezoid rule we can actually calculate the maximum error in the calculation to get an idea how much we are off of the actual value.

Trapezoid Error

To calculate the error when using the trapezoid rule, we use the following formula:

where K is the maximum value of |f’’(x)| on the interval.

3

2

( )| |

12T

K b aE

n

Midpoint Error

To calculate the error when using the midpoint rule, we use the following formula:

where K is the maximum value of |f’’(x)| on the interval.

3

2

( )| |

24M

K b aE

n

Example

Use n=6 to approximate the area under the curve on the interval [0, 1] using the midpoint rule.

2

( ) xf x e

Example

Calculate the error from the previous problem.

Example

What would you make n t0 ensure that you are within .0001 of the actual area

Example

Use n=6 to approximate the area under the curve on the interval [0, 1] using the trapezoid rule.

2

( ) xf x e

Example

Calculate the error from the previous problem.

Example

What would you make n to ensure that you are within .0001 of the actual area?

Homework

Approximate the area under the curve on the interval [0, 4] using both the midpoint and trapezoid rules. Find the error of both. N=6

2y x