Volume Under a Surface Over a Bounded Region

Find the volume under the surface $f(x, y) = 3x^{2} - 3xy^{2}$ over the region bounded by $y = x^{2}$ and $y = 2x$.

When calculating the volume under a surface defined by a function of two variables, it's essential to understand how to set up a double integral over the specified region. This involves identifying the boundaries of the region and determining the appropriate limits of integration. In this problem, the region is bounded by the curves $y = x^{2}$ and $y = 2x$, which form the bounds for the double integral. Understanding the interplay between these curves helps in accurately setting up the integral to compute the desired volume.

Additionally, selecting the correct order of integration can simplify the computation process. By analyzing the intersection points of the bounding curves, you can determine whether it's more efficient to integrate with respect to x first or y first. Mastery of these concepts is crucial for solving more complex problems involving volumes under surfaces and paves the way for tackling multidimensional integration challenges.

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