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Unformatted text preview: Lecture 22 Integration: Midpoint and Simpsons Rules Midpoint rule If we use the endpoints of the subintervals to approximate the integral, we run the risk that the values at the endpoints do not accurately represent the average value of the function on the subinterval. A point which is much more likely to be close to the average would be the midpoint of each subinterval. Using the midpoint in the sum is called the midpoint rule . On the i-th interval [ x i 1 ,x i ] we will call the midpoint x i , i.e. x i = x i 1 + x i 2 . If x i = x i- x i 1 is the length of each interval, then using midpoints to approximate the integral would give the formula: M n = n summationdisplay i =1 f ( x i ) x i . For even spacing, x = ( b- a ) /n , and the formula is: M n = b- a n n summationdisplay i =1 f ( x i ) = b- a n ( y 1 + y 2 + ... + y n ) , (22.1) where we define y i = f ( x i ). While the midpoint method is obviously better than L n or R n , it is not obvious that it is actually better than the trapezoid method T n , but it is. Simpsons rule Consider Figure 22.1. If f is not linear on a subinterval, then it can be seen that the errors for the midpoint and trapezoid rules behave in a very predictable way, they have opposite sign. For example, if the function is concave up then T n will be too high, while M n will be too low. Thus it makes sense that a better estimate would be to average T n and M n . However, in this case we can do better than a simple average. The error will be minimized if we use a weighted average. To find the proper weight we take advantage of the fact that for a quadratic function the errors EM n and ET n are exactly related by: | EM n | = 1 2 | ET n | ....
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This note was uploaded on 02/09/2012 for the course MATH 344 taught by Professor Young,t during the Fall '08 term at Ohio University- Athens.
- Fall '08