Problem 31 - Calculus of a Single Variable

Question

Finding the Area of a Region In Exercises \(31-36,\) (a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region analytically, and (c) use the integration capabilities of the graphing utility to verify your results. $$ f(x)=x\left(x^{2}-3 x+3\right), \quad g(x)=x^{2} $$

Step-by-Step Solution

Answer

The points of intersection are at \(x = 0, 1, 3\). The area between the curves is calculated by integrating the absolute difference of the two functions over these intervals. The final answer is confirmed using a graphing tool.

Step 1: Plot the functions

Use a graphing utility to sketch the graphs of the two functions. Identify the points of intersection, i.e., the values of \(x\) where the two curves cross each other.

Step 2: Find the points of intersection

Set the two functions equal to each other and solve for x to identify the interval over which the area is to be computed. Doing this will give: \(x(x^2 - 3x + 3) = x^2\). This simplifies as follows: \(x^3 - 3x^2 + 3x = x^2\) which further simplifies to \(x^3 - 4x^2 + 3x = 0\). On factoring this equation, you get \(x(x - 1)(x - 3) = 0\). So the points of intersection are \(x = 0, 1, 3\).

Step 3: Compute the area between the curves

This can be done using the formula to calculate the area between two curves: \(\int_{a}^{b} \left| f(x) - g(x) \right| dx\), where \(a\) and \(b\) are the points of intersection. To ease the calculation, two separate integrals are computed over the intervals [0,1] and [1,3]. In the first interval, \(g(x) > f(x)\), so the integral is \(\int_{0}^{1} (x^2 - x(x^2 - 3x + 3)) dx\). In the second interval, \(f(x) > g(x)\), so the integral is \(\int_{1}^{3} (x(x^2 - 3x + 3) - x^2) dx\). The required area is the sum of these two integrals.

Step 4: Verify your result

By graphing the two functions and the regions determined by them, the results from the integration can be verified with the integrated area function in a graphing utility.

Key Concepts

Area Between Curves

Calculating the area between two curves is a common problem in calculus. When two functions intersect, they create a region between them that can be measured using calculus. This task involves determining the area surrounded by two graphs in a specified interval. The general method involves:

• Identifying the points of intersection of the given curves.
• Setting appropriate integrations between these points.
• Subtracting the lower curve's equation from the upper one.

In the provided exercise, the curves are defined by the functions \( f(x) = x(x^2 - 3x + 3) \) and \( g(x) = x^2 \). The first step is finding their intersection points, which serve as the limits for our integral. After identifying these as \( x = 0, 1, \text{ and } 3 \), we can define two regions based on which function is greater over specific intervals to determine the total area.

Definite Integral

The definite integral is a powerful tool that helps calculate the area under a curve over a certain interval. It is denoted by \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) represent the interval limits, and \( f(x) \) is the function being integrated. In this exercise:

• The definite integral represents the sum of infinitely many infinitesimally small areas under the curve.
• The definite integral for the area calculation is split into two parts across two intervals that are bounded by the intersection points.
• A fundamental property of the definite integral allows us to use it to find areas even if the curves dip below the x-axis.

By using these properties, the definite integral allows us to accurately calculate areas between curves by subtracting functions over desired intervals.

Integration Techniques

Integration techniques are methods used to solve integrals which can often be tricky. For this problem, polynomial integration is key.

• Simplifying the expression \( f(x) - g(x) \) by expanding and combining like terms will often make solving easier.
• Once simplified, polynomial integration involves breaking down the polynomial into manageable parts through basic integration rules.
• When integrating \( \, \int x^n \, dx \), remember to use the formula: \( \frac{x^{n+1}}{n+1} + C \).

This method of breaking down complex polynomial expressions into simpler ones makes it straightforward to compute the area of each region between the curves in specified intervals of this problem.

Graphing Utility

A graphing utility is a helpful tool for visualizing functions and verifying mathematical calculations, particularly useful in calculus. Graphing functionalities include:

• Plotting the functions of interest to observe their interaction and relative positions.
• Identifying the intersection points visually to confirm algebraic solutions.
• Using integration features to confirm the calculated area between curves.

In this exercise, after calculating the area between the curves analytically, the graphing utility serves as a means to verify that the captured area matches across the computed intervals, thus reinforcing the correctness of the solution.