Problem 120 - University Calculus: Early Transcendentals

Question

In Exercises \(119-122,\) you will find the area between curves in the plane when you cannot find their points of intersection using simple algebra. 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). $$f(x)=\frac{x^{4}}{2}-3 x^{3}+10, \quad g(x)=8-12 x$$

Step-by-Step Solution

Answer

The total area between the curves is given by the sum of the integrals of \(|f(x) - g(x)|\) over each interval formed by consecutive intersection points.

Step 1: Plot the Curves

First, plot the functions \(f(x) = \frac{x^4}{2} - 3x^3 + 10\) and \(g(x) = 8 - 12x\) on the same graph. Use your computer algebra system (CAS) to graph these equations. This will help identify how many times the curves intersect and give a visual understanding of their behavior.

Step 2: Find Points of Intersection

Use the numerical equation solver in your CAS to find the points where \(f(x)\) and \(g(x)\) intersect. Solve \( \frac{x^4}{2} - 3x^3 + 10 = 8 - 12x \) to find values of \(x\) where the equations are equal.

Step 3: Set Up Absolute Value Integral

For each pair of intersection points \([a_i, b_i]\), set up the integral of \(|f(x) - g(x)|\). Use \(\int_{a_i}^{b_i} |f(x) - g(x)|\, dx\) for these regions.

Step 4: Compute Integrals

Use your CAS to calculate the definite integrals \(\int_{a_i}^{b_i} |f(x) - g(x)|\, dx\) for each region identified by the intersection points. Repeat this for every consecutive pair of intersection values.

Step 5: Sum the Integrals

Add the results of all integrals obtained in Step 4 to find the total area between the curves. This sum represents the total area between the curves \(f(x)\) and \(g(x)\) over all intervals where they intersect.

Key Concepts

Intersection Points

When dealing with curves in mathematics, finding the intersection points is crucial for various calculations, such as determining the area between two curves. Intersection points occur where the values of the curves are equal, essentially where they "meet" at a certain point on the graph. These points are important because they determine the boundaries of the region you will be analyzing or integrating over.

For the given functions, you will solve the equation:

• \[ \frac{x^4}{2} - 3x^3 + 10 = 8 - 12x \]

This equation cannot always be solved easily by hand due to its complexity, especially when they form higher degree polynomials or involve transcendental functions. Thus, numerical methods or computer-aided systems (like a Computer Algebra System) are often used to find these intersection points efficiently. Once the intersection points are determined, they act as limits for defining the integrals you'll ultimately solve.

Definite Integral

A definite integral is a type of integral that deals with finding the total accumulation of some quantity, usually area under a curve, over a specific interval. Unlike indefinite integrals, which result in a general formula, definite integrals yield a numerical value.

When finding the area between two curves, such as between

• \[ f(x) = \frac{x^4}{2} - 3x^3 + 10 \]
• \[ g(x) = 8 - 12x \]

we first determine the intersection points to establish the limits of integration. The integral is set up as:

• \[ \int_{a}^{b} |f(x) - g(x)| \, dx \]

The absolute value is crucial here because it ensures that the area is always positive, regardless of which curve is above the other over the specific interval. For each segment of the area between these intersection points, a separate integral is evaluated, and all these values are summed to find the total area between the curves.

Numerical Equation Solver

A numerical equation solver is a tool or algorithm used to find solutions to equations that may be too complex to solve analytically. These solvers use various numerical methods such as approximation techniques, iteration methods, or root-finding algorithms.

For the exercise given, a numerical equation solver will help solve

• \[ \frac{x^4}{2} - 3x^3 + 10 = 8 - 12x \]

By finding the values of \( x \) where this equation holds true, it identifies the points of intersection between the two curves. These are typically employed when dealing with nonlinear equations or higher degree polynomials, where traditional algebraic manipulation becomes inefficient or impossible. By leveraging a numerical solution, we can quickly and accurately find all necessary intersection points.

Computer Algebra System (CAS)

A Computer Algebra System, or CAS, is a software tool designed to carry out symbolic mathematical computations. In academic and professional settings, CAS is invaluable, especially when solving complex equations, plotting graphs, performing algebraic manipulations, or computing integrals.

In the context of finding the area between curves, a CAS can:

• Graph the functions accurately, providing a visual confirmation of the points of intersection.
• Utilize numerical solvers to find intersection points precisely, saving time and reducing the risk of errors compared to manual calculations.
• Calculate definite integrals, including those involving absolute values, with much greater speed and accuracy than by hand.

Using a CAS enhances the process by providing a clear and accurate picture, allowing students and professionals to focus on understanding the results rather than getting bogged down in tedious calculations.