Change integration limits for cylindrical to cartesian coord

[MCB277]

Homework Statement

I want to change the integration limits of an integral in cylindrical to cartesian coordinates. For example the integral of function f(r) evaluated between b and R: ∫ f(r)dr for r=b and r=R (there is no angular dependence).

For write de function in cartesian coordinates, use r=√(x^2+y^2) and rdr=dxdy, then, I should indicate an integration order for x and y.

Homework Equations

r=√(x^2+y^2)
∫ f(x,y)dx dy for x=? and y=?

The Attempt at a Solution

If I integrate in x first, de limit of integration should be x=-√(b^2-y^2) and x=-√(R^2-y^2), but for "y", what happens?.

Thanks

 

[Orodruin]

dr=dxdy

This is not correct. You cannot rewrite a one-variable integral into a two-variable integral like that. What you are looking for is $dx\,dy = r\, dr\, d\varphi$.