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 $$\int_{\varphi_1(\theta)}^{\varphi_2(\theta)}f(\rho\cos\theta,\rho\sin\theta)\rho d\rho ?$$