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Integration is a fundamental concept in calculus that involves finding the area under a curve. In the context of finding the area between curves, integration allows us to calculate the 'accumulated' area bounded by the curves on a certain interval.

Specifically, when the region between two curves is mentioned, we usually mean integration of the absolute difference between the two functions over the interval given. This is carried out by taking a series of infinitesimally thin vertical slices between the curves and summing their areas from one end of the interval to the other. Integration is useful not only for simple shapes but also for complex regions that cannot be easily computed using basic geometric formulas.

A linear function is one of the simplest forms of functions we have in mathematics. It is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.

Linear functions are graphically represented by straight lines and mathematically described by the formula: \( f(x) = mx + b \), where \( m \) represents the slope, and \( b \) represents the y-intercept. To graph a linear function, you can simply plot a couple of points that satisfy the equation and draw a line through them. The steepness of the line reflects the value of the slope, and where it crosses the y-axis indicates the y-intercept.

On the other hand, a quadratic function represents a parabola on the Cartesian plane. Its general form is \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is non-zero.

The graph of a quadratic function is a curve called a parabola that either opens upwards or downwards, depending on the sign of \( a \). The highest or lowest point of the parabola is known as the vertex, and it is a point of symmetry for the graph. Quadratic functions are very important in physics and other sciences, as they often describe the motion of objects under constant acceleration.

The definite integral of a function over an interval provides the net area between the function and the x-axis on that interval. The net area can be thought of as 'signed area', where the parts of the graph below the x-axis contribute negatively to the total.

When we speak of the definite integral, such as the one expressed by \( \int_{a}^{b} f(x)dx \), we are saying that we want to calculate the total area from \( x=a \) to \( x=b \) under the curve of \( f(x) \). In the context of our problem, where we have two functions, we take the definite integral of the difference between two functions to find the area enclosed by them, as demonstrated in the provided step-by-step solution. This is a powerful tool for understanding and calculating real-world quantities such as distances, areas, and volumes.