Module 19  Applications of Integration  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 19.1: Net Area  
In previous modules you used the definite integral to find the area bounded by a function and the xaxis. In each case the graph of the function was above the xaxis. In this lesson you will see what happens when the function dips below the xaxis. You will also investigate the concept of the definite integral as a netarea function. All the curves explored in Module 18 were above the xaxis. The following investigates a definite integral when part of the curve is below the xaxis.
How can the result be zero? The area bounded by y = sin x and the xaxis certainly is not zero. To help answer this question, break the interval of integration into two subintervals that represent the areas above and below the xaxis: [0, ] and [ , 2 ].
Finding Positive and Negative Integrals Review the graph of y = sin x and the values of the definite integrals and . The graph from 0 to is above the xaxis and the corresponding definite integral is positive. The graph from to 2 is below the xaxis and the corresponding definite integral is negative. Finding Net Area The definite integral represents the value of the net area, or the area above the xaxis minus the area below the xaxis. From the above, we know the area above the xaxis is 2 and the area below the xaxis is 2. The net area between the curve y = sinx and the xaxis on this interval is therefore 2 minus 2, or zero.
19.1.1 Use the definite integral feature in the CALC menu of the Graph screen to approximate the values of Click here for the answer. Visualizing You can obtain the general shape of a corresponding net area function on the interval [0, 2 ] by examining the graph of y = sinx.
The following characteristics of the net area function can be determined from the graph of the curve y = sinx. Look at the graph while reading the tables below. Recall that the curve function is the derivative of the net area function, or in this case, F ' (x) = sin(x).
Furthermore, the xvalues where maximums, minimums, and points of inflection occur can be identified by examining how the curve function is changing.
Other characteristics of the net area function include:
With these characteristics you can draw a graph of the net area function . Using the TI83 we can graph F(x) and support the above results. Graph F(x) by following the procedure below.
Visualizing the general shape of the integral function is often very helpful. Extending the Procedure to Other Curves 19.1.2 Graph the curve and find the net area bounded by y = x^{3} – 3x^{2} – x + 3 and the xaxis on the interval [0, 4]. Click here for the answer. Visualizing The characteristics of the net area function can be found by examining the graph of the curve function. The graph of the curve is shown below in a [0, 4, 1] x [ 5, 15, 1] window with a list of the net area function's characteristics.
The graphs of the curve function and the net area function are shown below for comparison.
19.1.3 Sketch the graph of the net area function that corresponds to the following curve in the window [0, 4, 1] x [2, 2, 1]. Click here for the answer. 

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