Math 135 Calculus I -
Lab 5 - Numerical Integration Methods I

 

We have seen two basic approaches for determining the value of a definite integral: the left or right hand Riemann sums. While the underlying ideas for each approach are the same, the obvious distinction between the methods gives us a means by which we can estimate integrals to an arbitrary level of precision - in most cases. Complications may arise when we try to "trap" the exact value between the left and right sums. In this lab you will be asked to investigate some of these complications. Our main tools will be Maple routines that are contained in the "student" package. Since these routines are contained in a package, Maple does not automatically load them when a session is started. You have to instruct Maple to do so:

> with(student);

Suppose we wish to estimate the value of 24x2dx using a left-hand Riemann sum with ten subdivisions. Since is an increasing function on [2,4], we would expect the left-hand sum to give us an underestimate for the true value of the integral. This observation is readily made if we use "leftbox" to generate a picture of the Riemann sum:

> leftbox(x^2,x=2..4,10);

[Maple Plot]

When using "leftbox" command (or any of the other three routines) we first specify the function, "x^2", then the interval, x=2..4, and finally the number of subdivisions. The sum is computed with "leftsum":

> leftsum(x^2,x=2..4,10);

[Maple Math]

Notice that Maple does not return a numerical value. It gives us a symbolic, or exact, value for the sum. We must tell Maple to give us a floating point (or decimal) answer by using the "evalf" command:

> evalf(%);

[Maple Math]

The percent sign in the evalf command is Maple shorthand that tells the computer to use the result of the preceding command. Rather than perform two separate commands, we can combine this into one command:

> evalf(leftsum(x^2,x=2..4,10));

[Maple Math]

Provided that the integrand is monotonic (increasing or decreasing) over the interval in question, we know that the exact value must lie somewhere between the left and right sums. If we increase the number of subdivisions to the point that the difference between these sums (in absolute value) is smaller than the amount of allowable error, we are assured that the error in our approximation is no greater than this difference. But suppose that the integrand is both increasing and decreasing over the interval in question, how do get an accurate estimate in this case?
Suppose we wish to determine the value of
with an error of less than .001, i.e., we want to be accurate to at least three decimal places. We could try to compare the left and right sums using, as an initial guess, say, five subdivisions:

> evalf(leftsum(exp(-x^2/2),x=-2..2,5));

[Maple Math]

> evalf(rightsum(exp(-x^2/2),x=-2..2,5));

[Maple Math]

Notice that the left and right sums yield identical values. Does this mean that we have found the exact value of the integral using only five subdivisions? Definitely not. This is obvious if we look at the sums with "leftbox" and "rightbox":

> leftbox(exp(-x^2/2),x=-2..2,5);

[Maple Plot]

> rightbox(exp(-x^2/2),x=-2..2,5);

[Maple Plot]

> f:=x->exp(-x^2/2);

[Maple Math]

Now Maple 'knows' of f as a function of x.

> evalf(f(-2));

[Maple Math]

Next we substitute the quantities into the expression and solve for n.  First, we ask Maple to evaluate {|f(-2)-f(0)|(2)}/n, then we ask it to solve to find when this expression is <.0001.

> evalf(abs(f(-2)-f(0))*2/n);

[Maple Math]

> solve(% <.0001);

[Maple Math]

Maple gives us two solutions, the numbers less than 0 and greater than 17293.29434.   Since we can't have a negative number of subdivisons, n must be greater than or equal to 17294. Notice that since the difference between the left and right sums is less than .0001 for this many subdivisions, we need only calculate one of the sums and we are guaranteed it will approximate the integral within .0001.

> evalf(leftsum(f(x),x=-2..0,17294));

[Maple Math]

Exercises
1. Compute the following integrals to two decimal places. For each integral, determine the minimum number of subdivisions required to achieve the desired level of accuracy.

a) [Maple Math] c) [Maple Math]
b) [Maple Math] d) [Maple Math]


2. Using the ideas detailed above, approximate the value of
[Maple Math] to at least three decimal places. It might be useful to look at a plot of the function (Remember the plot command: "plot(sin(x),x=0..3*Pi/2);").