MATH 134 -- Intensive Calculus for Science 2

Maple Numerical Integration Techniques Demo

February 13, 2001

Today we will see how Maple can be used to compute numerical

approximations to definite integrals using the LEFT, RIGHT,

TRAP, and MID methods. We will also see that Maple can compute

very accurate approximations using its own sophisticated built-in numerical

integration command.

Most of the commands we will use are contained in the "student" package.

To load them, we use:

> with(student);

For example, suppose we want to approximate

Here's a graphical display showing the left-hand sum for this function

with n = 10

> leftbox(sqrt(x^4+1),x=1..2,10);

LEFT(10) is computed by

> leftsum(sqrt(x^4+1),x=1..2,10);

The output is a short-hand form of the summation formula for the

left-hand sum. That's interesting, but we will want a numerical

value. To get a decimal approximation:

> ls:=evalf(leftsum(sqrt(x^4+1),x=1..2,10));

Similarly for RIGHT(10):

> rs:=evalf(rightsum(sqrt(x^4+1),x=1..2,10));

Similarly for MID(10):

> ms:=evalf(middlesum(sqrt(x^4+1),x=1..2,10));

Similarly for TRAP(10):

> tr:=evalf(trapezoid(sqrt(x^4+1),x=1..2,10));

Using Maple's sophisticated numerical integration routines:

> ex:=evalf(Int(sqrt(x^4+1),x=1..2));

The errors:

> errorleft:=ex - ls;

> errorright:=ex - rs;

> errortrap:=ex - tr;

> errormid:=ex - ms;

Repeating the computations for n = 100 -- LEFT(100):

> ls:=evalf(leftsum(sqrt(x^4+1),x=1..2,100));

RIGHT(100):

> rs:=evalf(rightsum(sqrt(x^4+1),x=1..2,100));

MID(100):

> ms:=evalf(middlesum(sqrt(x^4+1),x=1..2,100));

TRAP(100):

> tr:=evalf(trapezoid(sqrt(x^4+1),x=1..2,100));

Use the same exact value as before from Maple's sophisticated numerical integration routines:

The errors with n = 100:

> errorleft:=ex - ls;

> errorright:=ex - rs;

> errortrap:=ex - tr;

> errormid:=ex - ms;

What patterns do you notice?