Double integrals

>	restart; with(plots): with(VectorCalculus):

Warning, the name changecoords has been redefined

Warning, the assigned names <,> and <|> now have a global binding

Warning, these protected names have been redefined and unprotected: *, +, ., D, Vector, diff, int, limit, series

One way of representing a double integral is literally as two nested single integrals.  For example, over a rectangle:

>	I1:= int( 4*x-y^2, y=0..2 );

>	I2:= int( I1, x=0..3 );

If you use Int (capitalized), Maple will not evaluate the integral (the "inert" form).  This way you can see what is going on.

>	I1:= Int( 4*x-y^2, y=0..2 );

>	I2:= Int( I1, x=0..3 );

Use value to force Maple to do the evaluation.

>	value(I2);

To save time for all this, however, the VectorCalculus package defines a special syntax for int.

>	int( 4*x-y^2, [x,y]=Region(0..3,0..2) );

You can also ask for the "inert" form here.

>	int( 4*x-y^2, [x,y]=Region(0..3,0..2), 'inert' );

The integrals can be taken with variables in either order. Notice that the "outer" integral is listed first.

>	int( 4*x-y^2, [y,x]=Region(0..2,0..3), 'inert' );

Maple can also do integrals over nonrectangular regions.  Some are predefined for you.

>	int( x*y, [x,y]=Triangle(<0,0>,<2,0>,<2,1>) );

>	int( 1, [x,y]=Circle(<0,0>,4) );

More generally you need to work out variable limits on the "inner" integral.  For example, to integrate over this region:

>	implicitplot( {x=exp(y),x=2}, x=0..2, y=0..ln(2) );

We see that x varies from the value exp(y) to 2, as y varies from 0 to ln(2).  Hence we can use

>	int( f(x,y), [y,x]=Region(0..ln(2),exp(y)..2) );

An alternative is to do y first.

>	int( f(x,y), [x,y]=Region(1..2,0..ln(x)) );

They have to give the same answer!

>	eval(%,f(x,y)=x*y^2), eval(%%,f(x,y)=x*y^2);