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Polar coordinates are a useful way of describing points in a plane by their distance from a reference point (usually the origin) and their angle relative to a reference direction (usually the positive x-axis). This system is quite handy when working with circular or cylindrical shapes, as it simplifies many integral calculations.
In polar coordinates, any point \[ (x, y) \] is represented as \[ (r, \theta) \], where \[ r \] is the radial distance from the origin and \[ \theta \] is the angle. The relationships between Cartesian and polar coordinates are:

• \[ x = r \cos(\theta) \]
• \[ y = r \sin(\theta) \]
• \[ dA = r \, dr \, d\theta \]

Here, the area element \[ dA \] changes from \[ dx \, dy \] in Cartesian coordinates to \[ r \, dr \, d\theta \] in polar coordinates, which is crucial for setting up integrals in polar form.

A double integral extends the concept of an integral to functions of two variables, allowing you to compute areas, volumes, and other quantities over a 2-dimensional region. When dealing with functions of two variables, say \[ f(x, y) \], the double integral over a region \[ R \] in the \[ xy \]-plane is written as: \[ \iint_R f(x, y) \, dA \]
Here, \[ dA \] represents a differential area element. In Cartesian coordinates, this is simply \[ dx \, dy \], but in polar coordinates, it transforms to \[ r \, dr \, d\theta \], as discussed before.
To calculate the volume of a solid bounded by surfaces and a given region, you set up the double integral of the function representing the height of the solid over the region of interest. In our problem, this was the surface \[(x^{2} + y^{2} + 1)\] bounded by the region \[(x^2 + y^2 \leq 4)\] .

To find the volume of a solid using double integrals, follow these steps:

• Set up the integral: Identify the function that represents the height of the solid (in our case, \[(x^{2} + y^{2} + 1)\] ) and the region over which to integrate (here, \[(x^2 + y^2 \leq 4)\] ).
• Convert to polar coordinates: Translate the Cartesian coordinates into polar coordinates to simplify the integral, especially when the region involves circles or sectors.
• Evaluate the integrals: Integrate with respect to the radial distance first, and then with respect to the angle.

Specifically, for our example, we transformed the region \[(x^2 + y^2 \leq 4)\] into polar coordinates to get \[(0 \leq r \leq 2)\] and \[(0 \leq \theta \leq 2\pi)\]. We then evaluated the inner integral with respect to \[ r \] and the outer integral with respect to \[ \theta \], resulting in the final volume \[(12\pi)\].