Problem 93 - 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). $$ f(x)=x+\sin (2 x), \quad g(x)=x^{3} $$

Step-by-Step Solution

Answer

Visualize, find intersections, integrate absolute differences, and sum integrals.

Step 1: Visualize the Curves

Begin by plotting the curves of the functions \( f(x) = x + \sin(2x) \) and \( g(x) = x^3 \) using a computer algebra system (CAS). This will help you determine how they intersect and how many intersection points exist.

Step 2: Find Intersection Points

Use the numerical equation solver in a CAS to find the exact values of the points where \( f(x) = g(x) \), meaning \( x + \sin(2x) = x^3 \). Identify all points of intersection since algebraically solving this may be complex.

Step 3: Calculate Absolute Differences

For each consecutive pair of intersection points found in Step 2, compute the integral of the absolute difference between the functions: \( \int_{a}^{b} |f(x) - g(x)| \, dx \), where \( a \) and \( b \) are consecutive intersection points.

Step 4: Sum the Integrals

Add up all the integrals calculated in Step 3 to find the total area between the curves. This gives the complete area between the functions \( f(x) \) and \( g(x) \) across the range of interest.

Key Concepts

numerical equation solver

To find the points of intersection between complex functions, a numerical equation solver is invaluable. In this scenario, it helps find where \( f(x) = x + \sin(2x) \) intersects with \( g(x) = x^3 \). Using algebra alone can be challenging due to the transcendental nature of \( \sin(2x) \), so numerical solvers provide a practical alternative.

• These solvers use algorithms to approximate solutions to equations that are difficult or impossible to solve analytically.
• The algorithms iterate through possible solutions until they converge on a sufficiently accurate answer.

By inputting the equation \( x + \sin(2x) = x^3 \) into the solver, it iteratively computes values of \( x \) where both functions equal, identifying their intersections.

computer algebra system

A computer algebra system (CAS) is a powerful tool for visualizing and solving complex mathematical problems. In this exercise, it's used primarily for plotting the curves and employing numerical solvers.

• CAS provides a graphical view of how functions behave relative to each other, making it easier to identify intersection points visually.
• It aids in checking the nature of these points — whether they are tangents, crossings, or otherwise.
• The interactive environment allows for manipulation of equations in ways traditional algebra does not.

These systems support a deeper understanding by offering different angles through which one can analyze mathematical functions, especially those involving trigonometric elements like \( \sin(2x) \).

integration of absolute differences

Once the points of intersection are known, the next step is to understand the space between the curves. This involves calculating the integral of their absolute difference. The absolute value \( |f(x) - g(x)| \) ensures that the area measured is always positive, even if one function dips below the other.

• The integration spans from one intersection point to the next, ensuring that only the relevant sections of the curve contribute to the total area.
• This method provides an accurate representation of the enclosed space because it respects the aforementioned changes in dominance between the curves.
• Focus is on the magnitude of the difference rather than its direction.

The result of these integrals gives the area between the curves over each interval.

points of intersection calculation

Finding where two functions intersect is crucial for calculating the area between them. These points indicate limits for integration.

• First, the mathematical condition \( f(x) = g(x) \) must be satisfied, signifying that both functions equal at these points.
• Plotting them provides a visual representation that assists in solving the equation numerically.
• Each intersection divides the plane into distinct sections where one function is greater than the other.

Understanding these points lets us frame the sections of interest under the integral, effectively partitioning the area into manageable segments. Accurate calculation of these points ensures each segment is correctly accounted for in the integration process.