Problem 122 - 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). $$f(x)=x^{2} \cos x, \quad g(x)=x^{3}-x$$

Step-by-Step Solution

Answer

Use a CAS to find intersection points, then calculate and sum integrals of \(|f(x)-g(x)|\).

Step 1: Plot the Curves

To understand the nature of the curves and their intersections, plot the given functions: \( f(x) = x^2 \cos x \) and \( g(x) = x^3 - x \). Observe the graph to identify approximately how many points they appear to intersect. These points will guide us in the next steps for precise calculation.

Step 2: Find Points of Intersection

Use a numerical equation solver in your CAS (Computer Algebra System) to solve \( f(x) = g(x) \). This involves solving the equation \( x^2 \cos x = x^3 - x \) for \( x \). These solutions give precise intersection points needed for integration.

Step 3: Integrate the Absolute Difference

For each pair of consecutive intersection points \( a \) and \( b \) found, calculate the definite integral \( \int_{a}^{b} |f(x) - g(x)| \, dx \). This integral represents the area between the curves from \( a \) to \( b \). Repeat this process for each segment between intersections.

Step 4: Sum the Areas

Summarize the total area between the curves by adding up all the individual integrals calculated in Step 3. This sum represents the total area enclosed between the curves \( f(x) \) and \( g(x) \) over the ranges of intersection.

Key Concepts

Area between curves

Finding the area between curves is a fundamental concept in calculus. It involves determining the space sandwiched between two function graphs.
Imagine drawing two curves on a graph. The area between them is where these curves overlap vertically.
To calculate this area, we subtract the lower curve from the upper curve and take the integral of the absolute difference over a given interval.

• The smaller this interval, the more accurate the area calculation.
• Taking the absolute value ensures that area is always positive, no matter the orientation of the curves.
• The key step is identifying the correct intervals, which are defined by the points where the curves intersect.

By summing these integrals, you arrive at the total area between the curves.

Numerical equation solving

Numerical equation solving comes into play when traditional algebra fails to find intersection points. This technique uses computational power to approximate solutions.
You use a computer algebra system (CAS) to solve equations like \( f(x) = g(x) \), which can be complex by nature. Numerical methods approximate the solutions of these equations through iterative processes.

• Such methods might involve algorithms like the Newton-Raphson method or bisection method.
• Accuracy depends on computing power and the starting approximations provided.
• Mainly beneficial when dealing with transcendental functions or complex polynomials.

By utilizing these methods, you can find exact or nearly exact points where the curves intersect.

Definite integrals

A definite integral calculates the accumulation of quantities, such as area, over an interval \([a, b]\). In this context, we specifically look at the integral of the absolute difference between two functions.
Integrating provides a sum of infinite, minute quantities that together form a whole.

• Think of integration as adding up tiny slices under a curve, giving you total area or accumulated value.
• For the area between curves, it captures the space enclosed between each pair of intersection points.
• The definite integral has both lower and upper limits, indicating exactly where to start and stop.

Each calculated integral from intersections contributes to understanding the full-space coverage between the two curves.

Intersection points of functions

The intersection points of functions are the x-values where two functions meet on a graph. These points are crucial for determining the intervals over which to integrate.
When functions intersect, it means they have the same value at certain inputs.

• Simply put, solve \( f(x) = g(x) \) to find these points.
• These x-values act as boundaries for integration, defining where one function overtakes the other.
• Each pair of consecutive intersection points forms the segment of the graph over which you measure the area.

Identifying these points accurately is essential for calculating the area between curves effectively.