Easy Polar Coordinates question (Change of variables)

[EngineerHead]

I have a question regarding problem solving tips.

When given an iterated integral and asked to convert it to polar coordinates, how do you select the bounds of theta - do you have to understand how the graph of r operates and therefore know where the theta bounds are based on the rectangular coordinate bounds (aka completely conceptual)? Or is there a mathematical way of solving for the theta bounds?

For instance:
Set up a double integral in polar coordinates for:
f(x,y) = x+y
R: x^2 + y^2 ≤ 4
x ≥ 0
y ≥ 0

Obviously, the theta bounds are from 0->pi/2 because of the y and x bounds, but is there a mathematical procedure to arrive at this same answer? Or must you figure it out conceptually.

Thank you in advance for your help!

 

[LCKurtz]

You use the graph, perhaps with a little algebra to find intersection points. Just like in rectangular coordinates. You always (should!) draw the graph first.

 

[HallsofIvy]

Since x= r cos(\theta) the boundary line x= 0 corresponds to \theta= 0. Since y= r sin(\theta), the boundary line y= 0 corresponds to \theta = \pi/2. Since x^2+ y^2= r^2, the boundary x^2+ y^2= 1 gives r^2= 1 or r= 1 since r cannot be negative.