Evaluating a Double Integral with Polar Coordinates

Evaluate the double integral $\displaystyle \int_{0}^{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} e^{-r^2} r \, dr \, d\theta$ by converting it to polar coordinates in the region bounded by $x = \sqrt{4 - y^2}$ and the y-axis.

In this problem, we explore a fascinating application of integration in the context of polar coordinates, which often presents a more natural framework for tackling certain types of problems. The task of evaluating the given double integral involves converting a region described in Cartesian coordinates into polar coordinates, leveraging the symmetry and inherent characteristics of the polar system. By doing so, we can simplify the computation significantly.

The region of integration is bounded by the curve x equals the square root of 4 minus y squared and the y-axis. This type of problem invites us to think about the geometry of the situation, as it involves a semicircular region in the Cartesian plane. The use of polar coordinates becomes particularly useful here because of the circular symmetry of the region. In scenarios where the region or the functions involved are circular or radial in nature, polar coordinates can simplify the integration process by aligning our coordinate system with these characteristics.

Additionally, this problem serves as a great introduction to the technique of switching from one coordinate system to another. Understanding when and how to convert to polar coordinates is not only a valuable skill in calculus but also lays the groundwork for more advanced topics in mathematics and physics. Polar coordinates provide an efficient tool for handling problems with radial symmetry and can often turn complex integrals into more manageable forms.

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