If r is constant, dr/dΘ is 0, and the integral evaluates as πr^2 -- the circumference of a circle. This makes sense since the curve would be a circle if r is constant.

If r is not constant, then you have some other shape being described, and C is the length of the curve. You can think of it as the length an ant would walk if it followed the line of the graph of (r, f(Θ)).

eric_mathgeekIf r is constant, dr/dΘ is 0, and the integral evaluates as πr^2 -- the circumference of a circle. This makes sense since the curve would be a circle if r is constant.

If r is not constant, then you have some other shape being described, and C is the length of the curve. You can think of it as the length an ant would walk if it followed the line of the graph of (r, f(Θ)).