Volume Under a Paraboloid

Compute the volume under the surface given by $f(x, y) = 9 - x^2 - y^2$ over the rectangular region where $x$ is between $-2$ and $2$ and $y$ is between $-2$ and $2$.

To solve this problem, we need to compute the volume under a surface described by a function over a specified region in two-dimensional space. This is a classic application of double integrals in calculus, which allows us to find volumes of regions bounded by surfaces. The concept here involves understanding how to set up and compute a double integral over a rectangular region.

To begin with, note that the given surface is a paraboloid, which is symmetric and continuous over the defined region. The region of integration is a square in the xy-plane, specified by the limits provided for $x$ and $y$. One of the strategies is to express the function in terms of $x$ and $y$ and integrate it over this region. Since the problem involves integration of a function that describes a three-dimensional shape, we are essentially summing up infinitesimal volumes beneath the surface.

While setting up the double integral, one must be careful to respect the limits for each variable. It's often useful to sketch the region of integration to gain a visual understanding of the problem. Once the double integral is established, it becomes a matter of performing the integration with respect to $y$ and then $x$, or vice versa. Additionally, recognizing the symmetry in the function and region can simplify calculations. The symmetry about both axes suggests that the volume can be broken down into simpler, perhaps even identical, subproblems to reduce computational effort.

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