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When faced with the challenge of finding the area between two curves, one of the first and most crucial steps is to graph the functions. Graphing functions serves multiple purposes: it provides a visual representation of the functions themselves, it helps to identify the region whose area we are interested in, and it assists in finding the points of intersection between the functions.

In the context of our exercise, graphing the functions \(f(x) = x^{4}-9x^{2}\) and \(g(x) = x^{3}-9x\) allows us to visually capture the shape and behavior of the curves. Typically, we look for where the curves cross each other as well as the behavior at the endpoints to understand the bounds of our area of interest. Utilizing a graphing utility can simplify this step, enabling you to quickly see intersections and estimate regions.

To elaborate on the points of intersection, these are the x-values where the two curves meet. They are pivotal in our quest to determine the area between the curves, as they define the limits of integration.

Discovering these points analytically involves setting the two functions equal to each other, \(f(x) = g(x)\), and solving for x. This solution will yield the critical x-values. In our specific problem, solving \(x^{4}-9x^{2} = x^{3}-9x\) will provide us with the intersection points. Understanding the significance of these points and solving for them correctly is integral to the success of finding the area accurately.

The concept of the definite integral is at the heart of finding the area between two curves. A definite integral calculates the accumulation of quantities, which in our case, translates to the area under a curve. When we have two curves, the area between them is essentially the difference in their respective areas over the same interval.

Mathematically, this area can be expressed as \[\int_{a}^{b}|f(x) - g(x)| dx\], with 'a' and 'b' being the points of intersection. Why the absolute value? The absolute difference ensures that we always subtract the lower function from the upper function, regardless of which function is on top visually on the graph. Understanding how to set up and evaluate a definite integral is key to finding the correct area.

Dealing with integration, especially in the context of finding areas between curves, often requires a strong grasp of various integration techniques. These techniques can include substitution, integration by parts, trigonometric integration, and sometimes numerical integration methods, among others.

In our exercise, the integration may not be straightforward due to the absolute value and the polynomial nature of the functions. It's essential to decide whether to split the integral into sections or to use another technique to simplify the problem. Familiarity with integration techniques will assuredly impact the ease and swiftness with which one can find the area between the curves, reinforcing the importance of this concept in calculus.