# Double Integrals in Polar Coordinates and the geometric meaning

Let the general polar region $$D$$ be $$D=\{(\rho,\theta)\mid \alpha\leq\theta\leq\beta, \varphi_1(\theta)\leq \rho\leq\varphi_2(\theta)\}$$ and let $$f(x,y)$$ be a Riemann integrable function over $$D$$. Then the the double integral $$\iint_{D}f(x,y)dxdy$$ can be computed by $$\iint_{D}f(x,y)dxdy=\int_\alpha^\beta \left(\int_{\varphi_1(\theta)}^{\varphi_2(\theta)}f(\rho\cos\theta,\rho\sin\theta)\rho d\rho\right)d\theta.$$

My question is what is the geometric meaning of integral $$\int_{\varphi_1(\theta)}^{\varphi_2(\theta)}f(\rho\cos\theta,\rho\sin\theta)\rho d\rho ?$$

If $$f(x,y) = 1$$, it means the rate of change of area with respect to $$\theta$$, or $$\frac{dA}{d\theta}$$. This comes up for example in Kepler's Laws of Planetary motion, sweeping out equal areas in equal times, $$\frac{dA}{dt}$$ can be related if you know $$\frac{d\theta}{dt}$$.
In general it means the rate of change of the quantity getting integrated with respect to $$\theta$$. The meaning will depend on what sort of thing is actually being computed. If you are imagining a $$3D$$ shape where $$f(x,y)$$ represents the height, then the expression you asked about is the rate of change of volume with respect to theta, $$\frac{dV}{d\theta}$$.
• Thank you for your explanation. $\frac{d\theta}{dt}$ means the angular velocity?
A magic phrase that might be helpful here (or might be further confusing) is volume form. Just as the expression $$dx\ dy$$ expresses the area of an infinitesimally small rectangle in cartesian coordinates, the expression $$\rho\ d\rho\ d\theta$$ expresses the area of an infinitesimally small rectangle in polar coordinates. So what does this mean for the inner integral? Well, there are a couple of different ways of looking at it. One is that it's the usual area-under-the-curve, weighted by the relative 'length' of the infinitesimally small cross-section. It also has an interpretation from a physical perspective, though: it's the moment integral with respect to the central point (or axis) $$\rho=0$$