Problem 94 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Use the graphing utility of your calculator to draw the graphs of the curves \(y=-x^{3}-2 x^{2}+5 x-2\) and \(y=x \ln x\) on the same screen. Use ZOOM and TRACE or some other feature of your calculator to find where the curves intersect, and then compute the area of the region bounded by the curves.

Step-by-Step Solution

Answer

Graph the functions, find intersection points, and then integrate the difference of the two functions between these points to find the area.

Step 1 - Enter the Functions

Input the functions into the graphing utility. Enter the first function as \( y = -x^{3} - 2x^{2} + 5x - 2 \) and the second function as \( y = x \ln x \).

Step 2 - Graph the Functions

Use the ZOOM feature to adjust the viewing window so that both graphs are visible on the screen. Make sure to use an appropriate range for both the x-axis and y-axis.

Step 3 - Identify Intersection Points

Use the TRACE feature on your calculator to move along each curve and find the points where the two graphs intersect. Record these intersection points.

Step 4 - Set Up the Integral

Write the integral for the area between the curves based on the intersection points. The area \(A \) between the curves can be found using the formula: \[A = \int_{a}^{b} [y_{\text{upper}} - y_{\text{lower}}] \, dx \] where \( y_{\text{upper}} \) and \( y_{\text{lower}} \) are the equations of the curves and \( a \) and \( b \) are the x-coordinates of the intersection points.

Step 5 - Calculate the Area

Subtract the lower function from the upper function within the bounds given by the intersection points, and perform the integration. \[A = \int_{a}^{b} [(-x^{3} - 2x^{2} + 5x - 2) - (x \ln x)] \, dx. \] Compute this definite integral to find the area.

Key Concepts

Graphing Functions

When you need to find the area between two curves, the first step is to graph the functions. This helps in visualizing where the curves intersect and understanding the enclosed region.
To graph a function, input its equation into a graphing utility. Let's take the example of the functions given in the problem:
- The first function is: \( y = -x^{3} - 2x^{2} + 5x - 2 \)
- The second function is: \( y = x \ln x \)
After entering these equations into a graphing calculator, use the ZOOM feature to adjust the viewing window so both graphs are clearly visible. This ensures an appropriate range for both the x-axis and the y-axis, making the curves easier to analyze. Making the graphs visible helps in the next steps, which involve finding where these curves intersect.
The visualization from graphing the functions sets the stage for us to find intersections and subsequently compute the area between the curves.

Intersection Points

After graphing your functions, the next critical step is identifying the intersection points. Intersection points are where the two curves meet or cross each other. These points are essential because they define the limits (a and b) for setting up our integral.
To find the intersection points, use the TRACE feature on your graphing calculator. This tool allows you to move along each curve and observe the coordinates where they intersect. Keep a record of these points, specifically the x-coordinates. They will act as the bounds for the area calculation.
Without these points, you wouldn't know over which interval to integrate. For example, if the curves intersect at x = 1 and x = 4, then a = 1 and b = 4. These points also help in determining which function becomes the upper function (y_upper) and which one becomes the lower function (y_lower) in the area formula.

Definite Integral

Once the curves are graphed and the intersection points identified, the last step is computing the area between the curves using a definite integral.
A definite integral helps in finding the area under a curve from one point to another. For our problem, we are more interested in the area between two curves from one intersection point to another.
The area A between two curves can be calculated using the integral formula: \[ A = \int_{a}^{b} \left[ y_{\text{upper}} - y_{\text{lower}} \right] \ dx \] Here, \( y_{\text{upper}} \) is the function that lies above the other within the interval defined by the intersection points, while \( y_{\text{lower}} \) is the function that lies below. To find this area, subtract the lower function from the upper function and then integrate over the interval from a to b.
For the functions given in our example: \[ A = \int_{a}^{b} \left[ (-x^{3} - 2x^{2} + 5x - 2) - (x \ln x) \right] \ dx \] By computing this integral with your graphing calculator, you can find the exact area between the two curves within the specified limits.
Understanding these integral steps will help you solve similar problems efficiently and with confidence.