When our book first defined the definite integral,
defined it as signed area under a curve.
This was computed as the limit of an approximating sum (a Riemann sum),
usually the sum of the areas of a collection of rectangles. The
book then showed that the definite
integral was the limit of any Riemann sum as the number of subintervals
infinity and the size of the largest subinterval goes to zero.
When we use these sums to find an
approximation of a definite integral we say that we are doing numeric
integration, which is the main subject of chapter 7 of our
book. We will use a Java Applet to examine numeric
integration. To use the applet,
start Internet Explorer and point the browser to http://www.slu.edu/classes/maymk.
and select the Riemann Sums Applet.
1) The default example of the applet looks at numeric approximations to . The applet has 5 choices of approximation methods that can be graphed.
A) Use the double n and reset n buttons to determine how many intervals are needed with each method until it looks like a good approximation.
B) Type in your guess at the anti-derivative in the G(x) text field. Click the check box so that G(x)-G(xMin) is graphed. Adjust the limits on y to [-15,11] so that the graph of the anti-derivative appears. Click the box so that the numeric integral is graphed. Determine how many intervals are needed for the two graphs to be good approximations to each other.
C) How do we determine if we want G(x) or G(x)-G(xMin)?
2) Reset n and look at the area for .
A) Make a table of the values obtained by the 6 methods when n is 10, 40, 80, 320, 1280, and 81920. To what value do the methods converge? How does this value compare to the value given by numeric integration on your calculator?
B) How many intervals are needed by each method to get an approximation within 0.01? Within 0.0001?
3) The classic problem for numeric integration is area under the normal curve. SAT verbal and mathematics scores are supposed to fall in a normal distribution with a mean of 500 and a standard deviation of 100. Thus a score of n corresponds to being standard deviations above the mean. In terms of percentile, a score s standard deviations above the mean is at the ( )th percentile.
A) Use 10 boxes to approximate the percentile ranking of someone who scored a 650 on the mathematics test.
B) Estimate the percentile ranking of a score of 790. How many boxes are needed to be accurate to .01?
C) Redo the example above, replacing the integral given with the integral to find the percentile of a score of 700 on the SAT verbal test.
4) Use numeric integration to estimate the integral from exercise 15 on page 451. (This problem can be made easier by noting that the area to be measured is made up of 100 identical pieces of a width of 2π.)