Numerical Integration I

September 23, 2003

MME 523

 

Our goal is to explore the approximation of integrals by two methods: Riemann Sums and the Trapezoid Rule.  Time permitting we may look at Simpson’s Rule as well.  The intermediate goals are to better utilize Maple and to review some integration.

 

Part One

 

Your goal for next class is to take each problem and develop, using Maple, two tables  (you might want to use Excel for their actual creation). One table is for Riemann Sum results, the other for Trapezoidal Rule results. Each has two columns:  the amount of decimal place accuracy you have, and how many steps it took (n)  to achieve it.

You may use the int function in Maple to get the real value for comparison purposes (since Maple tries to produce a symbolic result, you may need to issue the  evalf(%);  function to get a floating point display.

 

Also of interest is the student library, which may be opened by issuing       >with(student);

 

 

The integrals are:

 

a)                     b)                     c)

 

 

                                                            d)  

 

Part Two

 

Your goal in this part is to use the error estimate to predict how many steps (n) will be needed to achieve a given accuracy with the Trapezoidal Rule.  The error estimate is:

 

                                | En| <  (b –a)3M/(12n2)

 

where  En  is the error in using n trapezoids,  a and b are the ends of the interval and M is an upper bound on the second derivative of  the function being integrated.

 

The simplest way to get M is to graph the second derivative and read M off the graph. If in doubt, round up.

 

 

a)              3 , 5, 10 places            b)      to  3 and 10 places

 

 

Part Three

 

Comment on the functions  leftbox, middlebox and rightbox   as demonstration tools for introducing secondary students to integration.