Problem 27 - Essential Calculus

Question

In Exercises \(27-34,\) (a) use a graphing utility to graph the region bounded by the graphs of the equations, \((b)\) find the area of the region, 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 steps involved in solving this problem are plotting the graphs, finding the intersection points, calculating the area between curves using integration and then verifying these results by using the integration feature of the graphing utility. The exact area and intersection points can only be determined once the equations are actually solved.

Step 1: Graph the functions

Firstly, use a graphing utility to graph the functions \(f(x) = x(x^2 - 3x + 3)\) and \(g(x) = x^2\). Visually identify the region that is bounded between these graphs.

Step 2: Find the intersection points

To calculate the area between the two graphs, we must find the points where the graphs intersect. This is done by setting the functions equal to each other, i.e. \(f(x) = g(x) \to x(x^2 - 3x + 3) = x^2\), solve this equation to find the points of intersections. These points will be the limits of the integration while calculating the area.

Step 3: Calculate the area

The area between the two curves is given by the absolute value of the integral of the difference between the two functions from the first intersection point to the second. Write this as an integral and calculate it.

Step 4: Verify the result

Finally, verify the calculated area by using the integration capabilities of the graphing utility. Enter the same integral used in Step 3 and confirm that the tool provides the same result.

Key Concepts

Graphing Utility

Imagine having a powerful visual aid that can sketch any function, pinpoint where they meet, and even shade the space in between—this is what a graphing utility can do. In our exercise, such a tool is indispensable. It takes the functions f(x) and g(x), and manifests them on screen, allowing us to observe the shape and intersection visually. Importantly, graphing utilities not only make it easier to comprehend where the functions intersect—which is crucial for finding our limits of integration—but they can also help in verifying solutions. This is because, as we toggle between these abstract equations and their graphical representations, the transition from a concrete visual picture to abstract calculation becomes much smoother.

Here's a tip to ensure accuracy when using a graphing utility: always check the window settings (the range of the x and y axes) to ensure that they are wide enough to encompass all the important features of the graph, including intersection points. This will help prevent any missing or misleading information that could lead to errors in the determination of intersecting points and hence, the calculation of the area.

Integration

The concept of integration is like a mathematical magic trick—it allows us to accumulate quantities, such as areas, to ascertain the whole from mere slices. When we talk about finding the area between curves, what we are essentially doing is integrating the “top” function minus the “bottom” function over the interval defined by their points of intersection. This gives us the net area between the curves.

To add a layer of clarity, consider integration as a way to tally up an infinite number of infinitesimally thin rectangles between these curves, allowing us to measure the space they enclose. In the exercise, after identifying the functions f(x) and g(x) and locating the boundaries where they intersect, we deploy integration to add up all those tiny areas between them to find the total area.

Definite Integral

The definite integral is the key to unlocking the actual measurement of the area between curves. With clear boundaries—provided by the intersection points of our functions—we can calculate a precise value representing the area. Think of it as drawing a line in the sand; the definite integral respects these prescribed limits, unlike its counterpart, the indefinite integral, which is more about the general formula for the area rather than its specific value.

In the process detailed in our step-by-step solution, we found the integral of the difference between f(x) and g(x) across the defined interval. This definite integral thus calculates the actual size of the area in question. Students should remember that precision is key here: accurate calculation of intersection points, correct identification of the upper and lower functions, and careful execution of the integration process, all play vital roles in arriving at the correct answer.