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). $$ f(x)=x+\sin (2 x), \quad g(x)=x^{3} $$

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).
$$
f(x)=x+\sin (2 x), \quad g(x)=x^{3}
$$

Key Concepts

Piecewise Integration and Summation

Piecewise integration involves breaking up the integration domain into segments defined by the intersection points of the functions. Each segment is integrated separately due to the potential change in relative positions of the functions. Summing these piecewise integrals then gives the total accumulated difference over the entire range, providing a comprehensive measure even when the functions switch order.

Integration of Absolute Value of the Difference

Integrating the absolute value of the difference between two functions over a specific interval provides a measure of the accumulated vertical distance between the curves. This technique is important when quantifying the total deviation or area between curves, especially when the functions cross, as the absolute value ensures that all differences are considered as positive contributions.

Graphical Analysis

Graphical analysis involves plotting multiple functions on the same set of axes to visually interpret their behavior and observe where they might intersect or diverge. This technique is useful for gaining an intuitive understanding of function behavior, identifying potential points of intersection, and assessing which regions may be significant for further analysis.

Numerical Equation Solving

Numerical equation solving refers to using computational methods within a Computer Algebra System (CAS) to find approximate solutions to equations that are difficult or impossible to solve exactly using analytical methods. When applied to finding intersection points, it helps identify the x-values where two functions take equal values by solving an equation derived from setting the functions equal.

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