Sec15.01IntegrationChecker.mws

__Computing multiple integrals__

Worksheet by Mike May, S. J.- maymk@slu.edu

**Setting up and checking integrals,**

**Chapter 15**

`> `
**restart;**

This worksheet is intended to give templates that a student might use in checking homework problems.

**Multiple integration**

Much of this chapter deals with problems where you need to evaluate a multiple integral. It is worthwhile to note the Maple syntax used for multiple integrals, so that you can use Maple to check your work,

We do multiple integrals in Maple by nesting the integration commands.

It is useful to note that the command Int (upper case i) only sets up the integral. (It is good practice to use both so that you see that the integral is set up properly, as well as computing its value.)

`> `
**Int(Int(x^2+y^2+x*y,x = 1..3), y=2..5);**

Int(int(x^2+y^2+x*y,x = 1..3), y=2..5);

int(int(x^2+y^2+x*y,x = 1..3), y=2..5);

Note that sometimes the integral is easier to evaluate in one order or the other.

`> `
**Int(Int(exp(x^2),x=y..1),y=0..1);**

Int(int(exp(x^2),x=y..1),y=0..1);

int(int(exp(x^2),x=y..1),y=0..1);

Int(Int(exp(x^2),y=0..x),x=0..1);

Int(int(exp(x^2),y=0..x),x=0..1);

int(int(exp(x^2),y=0..x),x=0..1);

**Plotting regions of integration**

On of the major problems with multiple integrals is getting the limits of integration correct. It is useful to plot the region to check the limits of integration. The following sections have blocks of code to visualize the region of integration.

**Slicing up and down**

We start with an integral of the form
. For that integral we are interested in the region with x between a and b, and y between the curves
and
.

`> `
**lowx := 1; highx := 4;**

lowcurve := x -> sqrt(x):

highcurve := x -> x:

lines := {}:

for i from 0 to 5 do

xval := lowx + i*(highx - lowx)/5:

lines := lines union

{[[xval, lowcurve(xval)],[xval, highcurve(xval)]]}:

od:

plot([highcurve(x), lowcurve(x), lines[1], lines[2], lines[3], lines[4], lines[5], lines[6]] , x= lowx..highx, color=[green, red, black, black, black, black, black, black], thickness=3);

**Slicing right and left**

We also want an example with the order of integration reversed, i.e., an integral of the form
. For that integral we are interested in the region with y between a and b, and x between the curves
and
.

`> `
**lowy := 1; highy := 4;**

leftcurve := y -> y^2;

rightcurve := y -> y;

xrange := 0..16:

lines := {}:

for i from 0 to 5 do

yval := lowy + i*(highy - lowy)/5:

lines := lines union

{[[leftcurve(yval), yval],[highcurve(yval), yval]]}:

od:

plot({[leftcurve(y), y, y=lowy..highy],

[rightcurve(y), y, y=lowy..highy]} union lines,

x = xrange);

**N**
**umeric integration**

Some integrals will choke even Maple if done symbolically. If we simply want the numeric value of the definite integral, we compose evalf with the inert integral Int.

`> `
**Int(sqrt(9*x^4 - 6*x^2 + 2), x=-1..1);**

`evalf(Int(sqrt(9*x^4 - 6*x^2 + 2), x=-1..1))`=evalf(Int(sqrt(9*x^4 - 6*x^2 + 2), x=-1..1));

Int(sqrt(9*x^4 - 6*x^2 + 2), x)= int(sqrt(9*x^4 - 6*x^2 + 2), x);

Int(sqrt(9*x^4 - 6*x^2 + 2), x=-1..1)=int(sqrt(9*x^4 - 6*x^2 + 2), x=-1..1);

`> `

`> `