I've been working on a homework set for Calc III, right now we're emphasizing double integration and polar integrals. I keep having problems conceptualizing where to actually create my region of integration (and subsequently the actual integrand)
Here is the most recent problem
$$ \text{Using polar coordinates, evaluate the integral}\\\text{which gives the area which lies in the first quadrant between the circles }\\ x^2 + y^2 = 196\text{ and } x^2 - 14x + y^2 = 0 $$
So I start by setting up my problem and I graph out the circles
I can see the area between the circles, and if I were to approach this in cartesian terms I would probably integrate y variable first from one equation to the second equation, and then the x variable from 0 to their intercept (which is probably 14 which I mentally deduced by the coordinates and radii, I haven't actually solved it). I also know I need to come up with something for the integrand, but I wouldn't even know what to do there.
So here's what my thinking is so far for polar coordinates. I believe that because it is just the change in r and the change in theta, I should be able to just do integrate the arclength like this:
$$ \int_{0}^{\frac{\pi}{4}} \int_{7}^{14}r\:dr\:d\theta\\ = \frac{147 \pi}{8} $$
From my inexperienced perspective, I should have the correct answer but my homework says it is not correct. What do I need to be doing differently?