Calculus 5.5

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5.5 Numerical Integration
Mt. Shasta, California
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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 dont 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.
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Actual area under curve
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Left-hand rectangular approximation
(too low)
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Right-hand rectangular approximation
(too high)
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Averaging the two
(too high)
1.25 error
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Averaging right and left rectangles gives us
trapezoids
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(still too high)
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This gives us a better approximation than either
left or right rectangles.
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Compare this with the Midpoint Rule
(too low)
0.625 error
The midpoint rule gives a closer approximation
than the trapezoidal rule, but in the opposite
direction.
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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!
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This weighted approximation gives us a closer
approximation than the midpoint or trapezoidal
rules.
Midpoint
Trapezoidal
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Example
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Simpsons 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.
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