536
Chapter 5 | Integration
Figure 5.21 The area above the axis and the area below the axis are equal, so the net signed area is zero.
Suppose we want to know how far the car travels overall, regardless of direction. In this case, we want to know the area between the curve and the x -axis, regardless of whether that area is above or below the axis. This is called the total area . Graphically, it is easiest to think of calculating total area by adding the areas above the axis and the areas below the axis (rather than subtracting the areas below the axis, as we did with net signed area). To accomplish this mathematically, we use the absolute value function. Thus, the total distance traveled by the car is ∫ 0 2 | 60 | dt + ∫ 2 5 |−40| dt = ∫ 0 2 60 dt + ∫ 2 5 40 dt =120+120 =240. Bringing these ideas together formally, we state the following definitions. ⎦ . Let A 1 represent the area between f ( x ) and the x -axis that lies above the axis and let A 2 represent the area between f ( x ) and the x -axis that lies below the axis. Then, the net signed area between f ( x ) and the x -axis is given by ∫ a b f ( x ) dx = A 1 − A 2 . The total area between f ( x ) and the x -axis is given by ∫ a b | f ( x ) | dx = A 1 + A 2 . Definition Let f ( x ) be an integrable function defined on an interval ⎡ ⎣ a , b ⎤
Example 5.10
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online