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# MATH 2.   Integration over various domains      21 October 1996
# 
# These examples follow the class notes.
# 
# These examples are simple enough that you should be able
# to do them by hand! I am using them purely to illustrate
# the approach with MAPLE.
# 
# First function:
# 
> f := x+2*y;

                                  f := x + 2 y
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#   integrate first with respect to y
# 
> g := int(f,y=x^2..2-x);

                                              3    4
                            g := - 2 x + 4 - x  - x
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#   and then with respect to x
> h := int(g,x=-2..1);

                                         243
                                    h := ---
                                          20
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# Second function:
# 
> f := 2*x*y;\


                                   f := 2 x y
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#   integrate first with respect to x
> g:=int(f,x=y^2-1..y+1);

                                      2       2     2
                          g := (y + 1)  y - (y  - 1)  y
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#   and then with respect to y
> h := int(g,y=-1..2);

                                    h := 27/4
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# Evaluate the second function in the opposite order,
# so integrate first with respect to y and then with respect to x.
> h:=int(int(f,y=-sqrt(x+1)..sqrt(x+1)),x=-1..0)
> + int(int(f,y=x-1..sqrt(x+1)),x=0..3);

                                    h := 27/4
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# Find the area of a region:
> f:=1;

                                     f := 1
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#   integrate first with respect to x
> g:= int(f,x=y^2-1..y+1);

                                               2
                                 g := y + 2 - y
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#   and then with respect to y
> h:=int(g,y=-1..2);

                                    h := 9/2
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