Use a CAS to perform the following steps: a. Plot the curves...

Use a CAS to perform the following steps: a. Plot the curves together to see what they look like and how many points of intersection they have. b. Use the numerical equation solver in your CAS to find all the points of intersection. c. Integrate $|f(x)-g(x)|$ over consecutive pairs of intersection values. d. Sum together the integrals found in part (c). $$ f(x)=x^{2} \cos x, \quad g(x)=x^{3}-x $$

Use a CAS to perform the following steps:
a. Plot the curves together to see what they look like and how many points of intersection they have.
b. Use the numerical equation solver in your CAS to find all the points of intersection.
c. Integrate $|f(x)-g(x)|$ over consecutive pairs of intersection values.
d. Sum together the integrals found in part (c).
$$
f(x)=x^{2} \cos x, \quad g(x)=x^{3}-x
$$

Key Concepts

Absolute Value Integration

Integration of the absolute value of a difference between functions is used to compute the total area between curves. By integrating the absolute difference, the technique ensures that all area contributions are positive, thereby accurately measuring the separation between the two curves across intervals.

Graphing Functions

Graphing functions involves plotting their curves to provide a visual understanding of their behavior. This helps in identifying key features such as intersections, relative positions, trends, and symmetry, which are essential initial steps in many exploratory analyses of functions.

Intersection of Curves

Finding the intersection points of two functions involves solving the equation where the functions are equal. These points serve as critical boundaries for further analysis, such as determining the limits of integration when calculating the area between curves.

Numerical Equation Solving

Numerical equation solving refers to using computational methods and algorithms, often available in computer algebra systems (CAS), to find approximate solutions to equations that may not be easily solvable using analytical techniques. This is particularly useful for non-linear equations or when exact solutions are unwieldy.

Definite Integration

Definite integration is the process of calculating the exact area under a curve between two specific points. It yields a numerical value that represents the accumulated total of the function values over an interval, which is foundational in calculating areas and solving accumulation problems.

Summation of Integrals

When the area between curves is segmented by multiple intersection points, the overall area can be computed by adding the absolute areas calculated over each segment. This summation ensures that the complete area between the curves is accounted for, regardless of changes in their relative positions across the domain.

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