Module 16  The Fundamental Theorem  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 16.2: Area Functions, A Visual Approach  
This lesson develops area functions using a visual approach by exploring the values of the areas under a curve when the righthand endpoint changes. The visualization is displayed by graphing a definite integral in which the right end point is variable. If the curve and the lefthand endpoint are given, an area function can be found for a righthand endpoint that varies. It's necessary to distinguish between the independent variables for the curve function and for the area function. If x is used to denote the variable righthand endpoint of the region, a second variable, t, should be used in the description of the curve function. Letting x represent the righthand endpoint, the area under the curve and above the taxis between t = 1 and t = x can be found by evaluating the definite integral . For example, if x = 3 the area under the curve between x = 1 and x = 3 is about 1.09861, as illustrated and evaluated in the screens below. 16.2.1 Put your calculator in Approximate mode or just press to get the decimal values, and use the definite integral key to find the areas under between 1 and the following values of x. Record your results in the table.
Click here for the answer. 16.2.2 Make a scatter plot of your results and find the logarithmic regression equation that best fits the data. Graph this equation together with the scatter plot. Click on scatter plot or regression equation to review the associated keystrokes. Click here for the answer. Based on this data, it appears that the area function, A(x), may be ln x. Graphing Integrals Graph together with the scatter plot.
The graph will be slow to appear because the calculator has to compute a new definite integral for each point it plots. You should see the area function develop point by point. Notice that the graph of A(x) appears to be the same as the graph of the ln x, the regression equation we found above.
16.2.3 Based on these graphs, predict the values of
.
16.2.4 Describe how the Fundamental theorem applies. Visualizing Suppose that you don't know a formula for the curve function but you have its graph. You can determine the shape of the area function by analyzing the graph of the curve. Below is the graph of on the interval from 1 to x that was developed earlier. Visualize the area under the graph of f being filled with thin rectangular sticks where the height of each stick is the yvalue of the curve function. Each successive rectangular stick is shorter than the prior one. The Riemann sum is growing more slowly with each step as you move to the right because the area of each successive rectangle is less than the one before, as shown below. Using the Fundamental Theorem The Fundamental Theorem states that the derivative of the area function is the curve function. That means that the curve function values correspond to slopes of tangent lines of the area function. Some of the characteristics of the area function that can be determined from the graph of the curve function are listed below.
16.2.5 Examine the graph of f(x) between x = 0 and x = 4 shown below. Approximate the graph of the corresponding area function on the interval from 0 to 4. Click here for the answer. 

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