MapleNeatExample.mws

Neat Example

Switching the Order of Integration in Polar Coordinates

We are looking at finding the area of the region bounded by the graphs of theta = 1  and theta = sin(r) .

>    Int(Int(r,r=0..arcsin(theta)),theta=0..1);

Int(Int(r,r = 0 .. arcsin(theta)),theta = 0 .. 1)

>    value(%);

int(1/2*arcsin(theta)^2,theta = 0 .. 1)

>    evalf(%);

.2337005501

Using the integration order above we can only approximate the area (Maple does a good job of approximating).

>    Int(Int(r,theta=sin(r)..1),r=0..Pi/2);

Int(Int(r,theta = sin(r) .. 1),r = 0 .. 1/2*Pi)

>    value(%);

1/8*Pi^2-1

>    evalf(%);

.233700550

Switching the order of integration allowed Maple to compute the exact area.

The region is graphed below.

>    with(plots):

>    rplot:=polarplot(arcsin(theta),theta=0..1,thickness=2):

>    thetaplot:=polarplot([t,1,t=0..Pi/2],thickness=2):

>    display(rplot,thetaplot);

[Maple Plot]

>