Download Integration in maple

Integration in maple
J. Gerlach
Last Revision: February 1, 2011
Antiderivatives
The command to find antiderivatives is int . You need to communicate the function (the integrand)
and the variable of integration. Here is an example: In order to find the antiderivative of
just enter
(1.1)
The syntax of the command is int( function, variable). The same result can be achieved with the
inert version of the integration command (two steps, Int followed by value(%))
(1.3)
or with right-clicking on the function
(1.4)
integrate w.r.t. x
(1.5)
or by using the icon on the left
(1.6)
Maple doesn't know every integral; when it gets too hard, maple will just repeat the question:
(1.7)
Constant of Integration
Maple omits the constant of integration C. This can sometimes lead to seemingly contradictory
results. Let's say that we want to integrate the function
Direct calculation yields
(1.1.1)
(1.1.2)
(1.1.3)
On the other hand, we could also expand the function before we integrate. This leads to the
following:
(1.1.4)
(1.1.5)
(1.1.6)
Thus we have found the two anti-derivatives Y and Z. They are not identical since
(1.1.7)
(1.1.8)
But this is not a contradiction, since we know that all antiderivatives of the same function differ
by a constant.
Variable of Integration
Suppose we want to integrate y =
. First we define the expression
xt
If we consider x to be the variable of integration, we find
(1.2.1)
(1.2.2)
while with t as variable we obtain
(1.2.3)
The answers are completely different, which means that we need to pay attention to the second
argument of the integration command (do we mean dx or dt?). If the variable of integration is
neither x nor t, say it is z, then we get
(1.2.4)
Definite Integrals
Definite integrals are evaluated just as easily; just add the range of the variable (lower and upper
limit) to the int command. For example, let's say we want to integrate
on the interval [0,
4]. We enter
(2.1)
Maple returns the result, but it doesn't display the integral very well. It may be more advantageous to
use a two step method (Int followed by value) instead
(2.2)
Clickable math can be used as well
(2.3)
integrate w.r.t. x
(2.4)
=
(2.5)
Here I used the "Definite Integral" option under "Constructions", and then the "Evaluate (from inert)"
command. Using the icon on the left we get
(2.6)
If you are interested in numerical results, you may use either the evalf command, or the
"Approximate" option when you right-click. Example:
0.9084218056
(2.7)
or
(2.8)
at 10 digits
0.9084218056
When maple can't find an explicit result for a definite integral (because the antiderivative may be
hard to find), numerical evaluation is a good option. Example:
(2.9)
Maple doesn't know how to proceed. Alternative:
(2.10)
0.8277841559
(2.11)
Variable Limits of Integration
The upper and lower limits in an integral may be variable themselves. Maple can handle this.
Example: We define a new function F(x) as
(2.1.1)
In the literature this function is known as the Fresnel Sine function. Maple recognizes this:
(2.1.2)
The derivative of F is
(2.1.3)
as expected from the Fundamental Theorem of Calculus.
Here is another illustration. We can construct an antiderivative of any function f(x) by defining it
as
Then F'(x) =f(x), as long as f is continuous. Example with
better choice)
and a=1 (a = 0 may be a
(2.1.4)
Check the derivative of F:
(2.1.5)
The explicit formula for F is
(2.1.6)
with derivative
(2.1.7)
Numerical Integration
Numerical integration is the approximate calculation of the value of a definite integral. This is useful
when the integrand is a complicated function without a simple anti-derivative. Evaluations of the
function at the left or right endpoint, or in the middle of a subinterval are special cases of Riemann
sums (Section 4.2 and 4.3) and means of numeric approximations of a definite integral. The
Trapezoidal Rule and Simpson's Rule (Section 4.6) are other options.
All of these are built-in functions of the student package. We illustrate all of these methods for the
integral of the function
on the interval from 0 to 2 and we compare the
numerical approximations to the exact value of the integral.
First restart and bring up the student package.
Now define the function once and for all
(3.1)
Next find the exact result for the definite integral and name it TrueResult.
86
15
(3.2)
5.733333333
(3.3)
at 10 digits
A display of the approximating rectangles can be done with the leftbox, rightbox or middlebox
command. For any of these instructions you need to communicate the function, the interval and the
number of subintervals. Let's look at right endpoints for 5, 10 and 20 subintervals:
5
4
3
2
1
0
5
4
3
2
1
0
0
1
x
2
0
1
x
2
5
4
3
2
1
0
0
1
x
2
It becomes evident that more subintervals will lead to better approximations of the area under the
curve.
The "box" commands display only, you need the "sum" commands to actually calculate the areas.
We illustrate this process for rightsums.
(3.4)
at 10 digits
6.584960000
(3.5)
The rightsum command just shows the sum which needs to be computed, and we have to follow it
with an evaluation command.All of this can be done in a single step (and we will use this in the
future) :
6.584960000
(3.6)
For more subintervals we get
6.146560000
5.936660000
The commands work just the same for leftsums or middlesums.
(3.8)
Now let us discuss the accuracy of approximation schemes, where we compare the approximations to
the exact value of the integral. We shall always use 10 subintervals in the comparisons, and the exact
value was computed before
86
15
5.733333333
As expected, right or left sums do not fare well:
(3.9)
6.146560000
(3.10)
0.413226667
(3.11)
5.346560000
(3.12)
(3.13)
The average of the two estimates should be better, especially since one estimate is too high and the
other is too low.
5.746560000
(3.14)
0.013226667
(3.15)
This is quite an improvement! The Trapezoidal Rule (details in class) will yield the same result:
5.746560000
(3.16)
For the Midpoint Rule the rectangles are always taken in the middle of an interval, as the graph
below illustrates
5
2
0
0
1
x
2
The accuracy of this method is on the same order as the Trapezoidal Rule (roughly speaking, it is
usually twice as good).
5.726760000
(3.17)
(3.18)
Finally, Simpson's Rule can be thought of as a weighted average of the Midpoint Rule and the
Trapezoidal Rule.
5.733360000
(3.19)
0.000026667
(3.20)
Again, notice a major improvement in accuracy! Simpson's Rule can be calculated directly with the
simpson command (we have to use n=20 to get the same result as above).
5.733359999
(this is just a slight rounding discrepancy).
For technical reason the number of subintervals in Simpson's rule must be even.
(3.21)