Perhaps if you posted a specific example, we could help you out?How to calculate, non-graphically, the new limits of integration when change of variables r done to a double integral?
I DID:Plz show me how u arrived at the new limits '' "
Those values are the angle of the straight line from the origin to each of the two points (b,c), (a,c).integral might be with [itex]\theta[/itex] between arctan(c/b) and arctan(c/a) to get the region between those two lower vertices (I am assuming that d-c> b- a. Otherwise you will "hit" the vertex (b,d) before (a,c).)
Now on each line between those, r must go from the lower line, y= c, to the vertical line x= b. In polar coordinates, that is [itex]r sin(\theta)= c[/itex] to [itex]r cos(\theta)= b[/itex]. That means that r varies from [itex]c/sin(\theta)[/itex] to [itex]b/cos(\theta)[/itex].