Problem 2 - Calculus Early Transcendentals

Question

Draw the graphs of two functions \(f\) and \(g\) that are continuous and intersect exactly three times on \((-\infty, \infty) .\) Explain how to use integration to find the area of the region bounded by the two curves.

Step-by-Step Solution

Answer

Answer: To find the area between two continuous functions that intersect each other, determine the intersection points of the functions, and then use integration with the principle of subtraction. The area can be computed by adding the absolute differences of the function values for the corresponding intervals between the intersection points.

Step 1: Create two continuous functions that intersect three times

Let's consider two functions \(f(x) = x^3 - x\) and \(g(x) = -x^2 + 1\). These functions are continuous on \((-\infty, \infty)\). We chose these functions because they have simple expressions and their graphs have clear intersection points.

Step 2: Find the intersection points

To find the intersection points of \(f\) and \(g\), we need to solve the equation \(f(x) = g(x)\). That is: \(x^3 - x = -x^2 + 1\) Rearrange the equation: \(x^3 + x^2 - x - 1 = 0\) Factor the equation: \((x + 1)(x^2 - x - 1) = 0\) Now we can find the roots, which are the intersection points. \(x = -1\) is one root. The quadratic part can be solved using the quadratic formula: \(x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\) \(x=\frac{1 \pm \sqrt{1+4}}{2} = \frac{1 \pm \sqrt{5}}{2}\) So, the three intersection points are \(x=-1\), \(x=\frac{1 - \sqrt{5}}{2}\), and \(x=\frac{1 + \sqrt{5}}{2}\).

Step 3: Visualize the graphs of f and g

Plot the graphs of the functions \(f(x) = x^3 - x\) and \(g(x) = -x^2 + 1\). You can use a graphing calculator or an online graphing tool like Desmos or GeoGebra to help you visualize the two functions. You can verify that these graphs intersect exactly three times at the previously found intersection points.

Step 4: Find the area between the curves

To find the area of the region bounded by the curves, we can use integration with the principle of subtraction. The area A can be found as follows: \(A = \int_{x_1}^{x_2} |f(x) - g(x)|dx + \int_{x_2}^{x_3} |g(x) - f(x)|dx\) Where \(x_1=\frac{1 - \sqrt{5}}{2}\), \(x_2=-1\), and \(x_3=\frac{1 + \sqrt{5}}{2}\) are the intersection points. First, evaluate the integrals separately: \(A_1 = \int_{x_1}^{x_2} (f(x) - g(x))dx\) \(A_2 = \int_{x_2}^{x_3} (g(x) - f(x))dx\) Then add the two areas together to find the total area: \(A = A_1 + A_2\) This will give you the area of the region bounded by the two functions \(f(x)\) and \(g(x)\).

Key Concepts

Continuous Functions

Continuous functions are a fundamental concept in calculus. A function is continuous if, for every point in its domain, the limit of the function as it approaches that point equals the function's value at that point.
This means the graph of the function will not have any breaks, jumps, or gaps. Some important properties of continuous functions are:

• Smooth curves without sharp corners.
• The ability to be drawn without lifting the pen from the paper.
• Continuous over an interval or the entire real line.

When dealing with continuous functions like in our problem—defined by the functions \(f(x) = x^3 - x\) and \(g(x) = -x^2 + 1\)—we know their graphs are smooth continuous lines.
This is important for defining their intersection points and studying the area they bound.

Intersection Points

Intersection points of two functions occur where their graphs meet. They represent the values of \(x\) where the functions are equal, \(f(x) = g(x)\). Finding these points is essential in identifying the regions of interest between the curves.To find intersection points, solve the equation \(f(x) = g(x)\), turning it into a problem of solving for the roots of \(f(x) - g(x) = 0\).
In our exercise:

• The equation \(x^3 - x + x^2 - 1 = 0\) was rearranged to \((x + 1)(x^2 - x - 1) = 0\).
• This helped identify the roots \(x = -1\) and solving the quadratic part with the quadratic formula gave \(x = \frac{1 \pm \sqrt{5}}{2}\).

With these \(x\)-values, we discovered the intersection points at \(x = -1\), \(x = \frac{1 - \sqrt{5}}{2}\), and \(x = \frac{1 + \sqrt{5}}{2}\).
Understanding these points allows us to analyze the spaces between the curves.

Bounded Area

The bounded area is the region enclosed between two curves. To find this, it's necessary to know how the curves are positioned relative to each other over specific intervals.
Identifying the intersection points allows us to define these intervals.Here's how you can find the bounded area:

• Determine which function lies above the other between each pair of intersection points.
• Use the principle of integration subtraction to subtract the lower function from the higher one.
• For visual clarity, plot the graphs to see which function dominates along different sections.

For example, between \(x_1 = \frac{1 - \sqrt{5}}{2}\) and \(x_2 = -1\), \(f(x)\) might be above \(g(x)\), while between \(x_2 = -1\) and \(x_3 = \frac{1 + \sqrt{5}}{2}\), \(g(x)\) may lie above \(f(x)\).
These intervals will dictate how you compute the area using integration.

Integration of Functions

Integration is a powerful tool to calculate areas bounded by curves. By finding the integral of a function between two points on the \(x\)-axis, you can determine the area under the curve.
This can also extend to the area between two curves, using subtraction and definite integration.The process involves:

• Evaluating integrals of the form \(\int_{a}^{b} |f(x) - g(x)| \) dx, where \(a\) and \(b\) are the intersection points.
• The absolute value \(|f(x) - g(x)|\) ensures you're always calculating positive area.
• Add the results from separate integrals over their respective intervals.

For our functions \(f(x)\) and \(g(x)\), we compute:
\[ A_1 = \int_{x_1}^{x_2} (f(x) - g(x)) dx \] and \[ A_2 = \int_{x_2}^{x_3} (g(x) - f(x)) dx \] which are then summed to get the total area \(A = A_1 + A_2\).
This method is essential for accurate calculation of areas between two curves.