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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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5.5 Numerical Integration

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5 5 numerical integration

Photo by Vickie Kelly, 1998

Greg Kelly, Hanford High School, Richland, Washington

5.5 Numerical Integration

Mt. Shasta, California


5 5 numerical integration

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.


5 5 numerical integration

Actual area under curve:


5 5 numerical integration

Approximate area:

Left-hand rectangular approximation:

(too low)


5 5 numerical integration

Approximate area:

Right-hand rectangular approximation:

(too high)


5 5 numerical integration

Averaging the two:

(too high)

1.25% error


5 5 numerical integration

Averaging right and left rectangles gives us trapezoids:


5 5 numerical integration

(still too high)


5 5 numerical integration

Trapezoidal Rule:

( h = width of subinterval )

This gives us a better approximation than either left or right rectangles.


5 5 numerical integration

Approximate area:

Compare this with the Midpoint Rule:

0.625% error

(too low)

The midpoint rule gives a closer approximation than the trapezoidal rule, but in the opposite direction.


5 5 numerical integration

Trapezoidal Rule:

(too high)

1.25% error

Midpoint Rule:

0.625% error

(too low)

Ahhh!

Oooh!

Wow!

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!


5 5 numerical integration

twice midpoint

trapezoidal

This weighted approximation gives us a closer approximation than the midpoint or trapezoidal rules.

Midpoint:

Trapezoidal:


5 5 numerical integration

Simpson’s Rule:

( h = width of subinterval, n must be even )

Example:


5 5 numerical integration

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.

p


5 5 numerical integration

Simpson’s Rule:

Trapezoidal Rule:

( h = width of subinterval, n must be even )

( h = width of subinterval )

Use the 2 rules above

to estimate the integral

to the right:

Your Turn!

Then determine the error of each method by comparing to

the exact integral value calculated by the antiderivative.


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