Photo by Vickie Kelly, 1998. Greg Kelly, Hanford High School, Richland, Washington. 5.5 Numerical Integration. Mt. Shasta, California. Using integrals to find area works extremely well as long as we can find the antiderivative of the function.
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Greg Kelly, Hanford High School, Richland, Washington
5.5 Numerical Integration
Mt. Shasta, California
Using integrals to find area works extremely well as long as we can find the antiderivative of the function.
Sometimes, the function is too complicated to find the antiderivative.
At other times, we don’t even have a function, but only measurements taken from real life.
What we need is an efficient method to estimate area when we can not find the antiderivative.
Left-hand rectangular approximation:
Right-hand rectangular approximation:
( h = width of subinterval )
This gives us a better approximation than either left or right rectangles.
Compare this with the Midpoint Rule:
The midpoint rule gives a closer approximation than the trapezoidal rule, but in the opposite direction.
Notice that the trapezoidal rule gives us an answer that has twice as much error as the midpoint rule, but in the opposite direction.
If we use a weighted average:
This is the exact answer!
This weighted approximation gives us a closer approximation than the midpoint or trapezoidal rules.
( h = width of subinterval, n must be even )
Simpson’s rule can also be interpreted as fitting parabolas to sections of the curve, which is why this example came out exactly.
Simpson’s rule will usually give a very good approximation with relatively few subintervals.
It is especially useful when we have no equation and the data points are determined experimentally.