(a) use a graphing utility to graph the region bounded by...

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results.
$$
f(x)=x^{4}-4 x^{2}, g(x)=x^{2}-4
$$

Key Concepts

Graphing Functions

Graphing functions involves plotting equations on the coordinate plane to visually represent the behavior of algebraic expressions. This is essential in understanding how different functions interact, particularly their points of intersection and the overall structure of the region they enclose.

Intersection of Functions

Finding the intersection of functions is the process of solving for the x-values at which the outputs of two functions are equal. The intersection points are critical in determining the limits of integration when calculating the area between curves, as they mark where the roles of the functions change relative to one another.

Area Between Curves

The area between curves refers to the region enclosed by two graphical representations of functions. It is computed by integrating the difference between the top (upper) function and the bottom (lower) function over a specified interval, typically determined by the points of intersection.

Definite Integration

Definite integration is a fundamental concept in calculus that computes the net area under a curve between two specific points on the x-axis. It is used to calculate areas, among other applications, by evaluating the integral of a function within set boundaries.

Verification Using Graphing Utility

Utilizing a graphing utility allows for both the graphical visualization of functions and numerical or symbolic integration to verify analytic results. This verification process helps ensure that the computed area corresponds accurately with the visual representation of the bounded region.

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