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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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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.

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.


Actual area under curve: we can find the antiderivative of the function.


Approximate area: we can find the antiderivative of the function.

Left-hand rectangular approximation:

(too low)


Approximate area: we can find the antiderivative of the function.

Right-hand rectangular approximation:

(too high)


Averaging the two: we can find the antiderivative of the function.

(too high)

1.25% error


Averaging right and left rectangles gives us trapezoids: we can find the antiderivative of the function.


(still too high) we can find the antiderivative of the function.


Trapezoidal Rule: we can find the antiderivative of the function.

( h = width of subinterval )

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


Approximate area: we can find the antiderivative of the function.

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.


Trapezoidal Rule: we can find the antiderivative of the function.

(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!


twice midpoint we can find the antiderivative of the function.

trapezoidal

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

Midpoint:

Trapezoidal:


Simpson’s Rule: we can find the antiderivative of the function.

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

Example:


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


Simpson’s Rule: parabolas to sections of the curve, which is why this example came out exactly.

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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