Aerospace Radar Tracking Coverage Sector Area (Aerospace Engineering)

An aerospace radar systems engineer at the UNLV Howard R. Hughes College of Engineering is configuring an automated airspace surveillance phased-array radar unit at Harry Reid International Airport in Las Vegas.

The primary radar is calibrated to filter out near-field ground clutter, establishing an inner blind-zone radius of $r_{\text{inner}} = 10\text{ km}$, while its transmitter power budget limits solid target detection to a maximum instrumental range of $r_{\text{outer}} = 80\text{ km}$. To maintain high tracking data-rate updates along a critical high-density arrival corridor, the antenna electronically steers its beam across an azimuth swath from $\theta_1 = \frac{\pi}{6}\text{ rad}$ ($30^\circ$) to $\theta_2 = \frac{\pi}{3}\text{ rad}$ ($60^\circ$).

To determine sensor compute requirements and allocate digital track-processing channels:

1. Express the geographic tracking corridor as a bounded polar rectangle in terms of radial coordinates $r$ and angular sweep $\theta$.
2. Formulate the double integral representing the total geographic surface area monitored by the radar system.
3. Evaluate the double integral to determine the exact area in square kilometers ($\text{km}^2$).
4. Explain how spatial coverage calculations allow aerospace engineers to predict maximum simultaneous track load and eliminate surveillance coverage gaps between adjacent regional air-traffic control centers.