Problem 91 - Thomas Calculus

Question

In Exercises \(91-94\) , 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). $$ f(x)=\frac{x^{3}}{3}-\frac{x^{2}}{2}-2 x+\frac{1}{3}, \quad g(x)=x-1 $$

Step-by-Step Solution

Answer

Find intersection points of the curves and sum the areas between them.

Step 1: Plot the Curves

To begin, we need to plot the two functions: \( f(x) = \frac{x^3}{3} - \frac{x^2}{2} - 2x + \frac{1}{3} \) and \( g(x) = x - 1 \). This visual representation will help us see where the curves intersect and how many intersection points there are. You can use graphing software or a CAS to create these plots.

Step 2: Find Intersection Points

Using the numerical equation solver in a CAS, solve for the values of \( x \) where \( f(x) = g(x) \). This step involves setting the equation \( \frac{x^3}{3} - \frac{x^2}{2} - 2x + \frac{1}{3} = x - 1 \) and finding the solutions for \( x \). These solutions are the intersection points.

Step 3: Compute the Absolute Difference

For each interval determined by the intersection points found in Step 2, compute the integrand \( |f(x) - g(x)| \). The absolute value is important because the area between the curves must be positive.

Step 4: Integrate Over Intervals

Integrate \( |f(x) - g(x)| \) over each interval between pairs of consecutive intersection points. This step calculates the area between the two curves for each segment.

Step 5: Sum the Integrals

Add all of the integrals from Step 4 together to find the total area between the curves across all intervals. This will give the total area between \( f(x) \) and \( g(x) \).

Key Concepts

Intersection Points

In algebra and calculus, understanding where two functions intersect is crucial. Intersection points help us determine where two curves meet on a graph. In this particular exercise, we need to find where the function \( f(x) = \frac{x^3}{3} - \frac{x^2}{2} - 2x + \frac{1}{3} \) crosses the line \( g(x) = x - 1 \). By setting \( f(x) = g(x) \), we create an equation to solve: \[ \frac{x^3}{3} - \frac{x^2}{2} - 2x + \frac{1}{3} = x - 1. \] Solving this equation will give us the \( x \)-values where these two curves intersect. This step is vital because these \( x \)-values define the intervals we'll use to find the areas between the curves.

Numerical Equation Solver

Using a numerical equation solver is a practical approach when dealing with complex functions where finding intersection points algebraically is challenging. These solvers, often found within graphing calculators or computer algebra systems (CAS), approximate solutions to equations that are difficult to handle by hand. Rather than relying on straightforward algebraic manipulation, the solver utilizes iterative methods to calculate precise approximations for the \( x \)-values where \( f(x) = g(x) \). This allows us to accurately determine the intersection points without extensive manual calculations. By using technology, we can streamline our workflow and ensure our answers are precise.

Absolute Value

Understanding absolute value is essential when calculating the area between curves. This is because the area must always be positive. When we compute \(|f(x) - g(x)|\), the absolute value ensures that we take the positive difference between the functions \( f(x) \) and \( g(x) \) over the specified \( x \)-intervals. Think of absolute value as the 'distance' measurement between two points, disregarding direction. This means, even if one function dips below another, the number remains positive, accurately reflecting the true separation distance between the curves on a graph.

Definite Integral

The definite integral is the tool we use to find the area in calculus. Once we have the intersection points, they segment our domain into intervals. To find the area between \( f(x) \) and \( g(x) \), we compute the integral of \(|f(x) - g(x)|\) over each of these intervals. Mathematically, this is expressed as: \[ \int_{a}^{b} |f(x) - g(x)| \, dx, \] where \( a \) and \( b \) are consecutive intersection points. Integrating in this way for each interval gives us the area for that segment. After finding these integrals for all intervals, summing them provides the total area between the two curves. This method is effective in capturing the entire region, regardless of how the curves might loop or cross each other.