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Photo by Vickie Kelly, 1998. Greg Kelly, Hanford High School, Richland, Washington. Trapezoidal Rule. 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

Trapezoidal Rule

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 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.


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


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

Left-hand rectangular approximation:

(too low)


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

Right-hand rectangular approximation:

(too high)


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

(too high)

1.25% error


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. we can find the antiderivative of the function.

The formula we used in geometry to find the area of a trapezoid is:


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The area under the curve we can find the antiderivative of the function.f1(x) is a trapezoid. The integral

.

NOTE: We used only one trapezoid in this example.


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Trapezoidal Approximation we can find the antiderivative of the function.

Averaging right and left rectangles gives us trapezoids:


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Area of each trapezoid based on we can find the antiderivative of the function.

(still too high)


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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.


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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.


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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!


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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:


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