The net area function is concave downward on (0, 2.1547) because the curve is decreasing there. The net area function is decreasing on (1, 3) because the curve function is negative there. The net area function is increasing on (0, 1) and (3, 4) because the curve function is positive there. The graph of the curve is shown below in a x window with a list of the net area function's characteristics. The characteristics of the net area functionĬan be found by examining the graph of the curve function. Visualizing the general shape of the integral functionġ9.1.2 Graph the curve and find the net area bounded by y = x 3 3 x 2 x + 3 and the x-axis on the interval. Graph F( x) by following the procedure below. Using the TI-83 we can graph F( x) and support the above results. With these characteristics you can draw a graph of the net area function ] is 0, so the net area function is 0 when x = 2 ] is 2 and the net area function begins to decrease at that point, so the maximum of the net area function is 2 when x = Other characteristics of the net area function include: Recall that the curve function is the derivative of the net area function, or in this case, F ' ( x) = sin( x).įurthermore, the x-values where maximums, minimums, and points of inflection occur can be identified by examining how the curve function is changing. Look at the graph while reading the tables below. The following characteristics of the net area functionĬan be determined from the graph of the curve y = sin x. Refresh the graph of y = sin x by pressing You can obtain the general shape of a corresponding net area function on the interval [0, 2 In the CALC menu of the Graph screen to approximate the values of The net area between the curve y = sin x and the x-axis on this interval is therefore 2 minus 2, or zero. From the above, we know the area above the x-axis is 2 and the area below the x-axis is 2. Represents the value of the net area, or the area above the x-axis minus the area below the x-axis. Is below the x-axis and the corresponding definite integral is negative. Is above the x-axis and the corresponding definite integral is positive. Review the graph of y = sin x and the values of the definite integrals To help answer this question, break the interval of integration into two subintervals that represent the areas above and below the x-axis: [0, How can the result be zero? The area bounded by y = sin x and the x-axis certainly is not zero. The following investigates a definite integral when part of the curve is below the x-axis. You will also investigate the concept of the definite integral as a net-area function.Īll the curves explored in Module 18 were above the x-axis. In this lesson you will see what happens when the function dips below the x-axis. In each case the graph of the function was above the x-axis. In previous modules you used the definite integral to find the area bounded by a function and the x-axis. Module 19 - Applications of Integration - Lesson 1 Module 19 - Applications of Integration
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