Example 1 :
Change of order of integration in double integral
Let the compare the given integral to the integral in standard form
The bounding curves are given by
Let us plot these curves to get the region of integration
After changing the order, the integral will be of the form
To get the limit of let us draw a circular arc with centre at the origin oriented in anticlockwise direction.
The arc intersects first and then intersect . So, limit of is from to . To find out limit of r, we must find out in what range the radius of arc should vary so that whole region of integration is transerved. r should vary from r = 0 to r = 2a.
So, after changing the order, the integral changes to
Example 2 :
Change the order of integration in the system of integrals
Comparing the integral with its standard form
The region of integration is bounded by curves
After changing the order the integral is of the form
To find the limit of , let us draw an circular arc of radius r intersecting the region of integration arbitrary. The arc has two possibilities. If radius of arc is less than a it intersects circle and cardiod and if a < r < 2a, it intersects initial line and cardiod. So, we have to partition the domain of integration by a partitioning curve in the form of circular arc of radius a into region I and II.
For region, I let us draw a circular arc intersecting the region I arbitrary. The limit of to and r will vary from r = 0 to r = a.
For region II, let us draw a circular arc intersection the region II arbitrary. The limit of is from to and r will vary from r = a to r = 2a.
So after changing the order, the integral will convert to