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The Trapezoid Rule Help (page 2)

By — McGraw-Hill Professional
Updated on Sep 1, 2011

Example 1

Calculate the integral

Applications of the Integral 8.7 Numerical Methods of Integration

to two decimal places of accuracy.

Solution 1

We first calculate that if f ( x ) = e x 2 then f″ ( x ) = (4 x 2 − 2) e x 2 and therefore | f″ ( x )| ≤ 2 = M for 0 ≤ x ≤ 1. In order to control the error, and to have two decimal places of accuracy, we need to have

Applications of the Integral 8.7 Numerical Methods of Integration

Rearranging this inequality gives

Applications of the Integral 8.7 Numerical Methods of Integration

Obviously k = 6 will do.

So we will use the partition P = {0, 1/6, 1/3, 1/2, 2/3, 5/6, 1}. The corresponding trapezoidal sum is

Applications of the Integral 8.7 Numerical Methods of Integration

Some tedious but feasible calculation yields then that

Applications of the Integral 8.7 Numerical Methods of Integration

We may use a computer algebra utility like Mathematica or Maple to calculate the integral exactly (to six decimal places) to equal 0.746824. We thus see that the answer we obtained with the Trapezoid Rule is certainly accurate to two decimal places. It is not accurate to three decimal places.

It should be noted that Maple and Mathematica both use numerical techniques, like the ones being developed in this section, to calculate integrals. So our calculations merely emulate what these computer algebra utilities do so swiftly and so well.

You Try It : How fine a partition would we have needed to use if we wanted four decimal places of accuracy in the last example? If you have some facility with a computer, use the Trapezoid Rule with that partition and confirm that your answer agrees with Mathematica’s answer to four decimal places.

Example 2

Use the Trapezoid Rule with k = 4 to estimate

Applications of the Integral 8.7 Numerical Methods of Integration

Solution 2

Of course we could calculate this integral precisely by hand, but the point here is to get some practice with the Trapezoid Rule. We calculate

Applications of the Integral 8.7 Numerical Methods of Integration

A bit of calculation reveals that

Applications of the Integral 8.7 Numerical Methods of Integration

Now if we take f (x) = 1/(1 + x 2 ) then f″ ( x ) = (6 x 2 − 2)/(1 + x 2 ) 3. Thus, on the interval [0, 1], we have that | f″ ( x )| ≤ 4 = M . Thus the error estimate for the Trapezoid Rule predicts accuracy of

Applications of the Integral 8.7 Numerical Methods of Integration

This suggests accuracy of one decimal place.

Now we know that the true and exact value of the integral is arctan 1 ≈ 0.78539816.... Thus our Trapezoid Rule approximation is good to one, and nearly to two, decimal places—better than predicted.

Find practice problems and solutions for these concepts at: Applications of the Integral Practice Test.

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