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Chapter 6 | Applications of Integration
Introduction The Hoover Dam is an engineering marvel. When Lake Mead, the reservoir behind the dam, is full, the dam withstands a great deal of force. However, water levels in the lake vary considerably as a result of droughts and varying water demands. Later in this chapter, we use definite integrals to calculate the force exerted on the dam when the reservoir is full and we examine how changing water levels affect that force (see Example 6.28 ). Hydrostatic force is only one of the many applications of definite integrals we explore in this chapter. From geometric applications such as surface area and volume, to physical applications such as mass and work, to growth and decay models, definite integrals are a powerful tool to help us understand and model the world around us. 6.1 | Areas between Curves Learning Objectives 6.1.1 Determine the area of a region between two curves by integrating with respect to the independent variable. 6.1.2 Find the area of a compound region. 6.1.3 Determine the area of a region between two curves by integrating with respect to the dependent variable. In Introduction to Integration , we developed the concept of the definite integral to calculate the area below a curve on a given interval. In this section, we expand that idea to calculate the area of more complex regions. We start by finding the area between two curves that are functions of x , beginning with the simple case in which one function value is always greater than the other. We then look at cases when the graphs of the functions cross. Last, we consider how to calculate the area between two curves that are functions of y .
Area of a Region between Two Curves Let f ( x ) and g ( x ) be continuous functions over an interval ⎡ ⎣ a , b ⎤ area between the graphs of the functions, as shown in the following figure.
⎦ such that f ( x ) ≥ g ( x ) on ⎡
⎤ ⎦ . We want to find the
⎣ a , b
Figure 6.2 The area between the graphs of two functions, f ( x ) and g ( x ), on the interval [ a , b ].
As we did before, we are going to partition the interval on the x -axis and approximate the area between the graphs of the functions with rectangles. So, for i =0, 1, 2,…, n , let P ={ x i } be a regular partition of ⎡ ⎣ a , b ⎤ ⎦ . Then, for i =1, 2,…, n , choose a point x i * ∈ [ x i −1 , x i ], and on each interval [ x i −1 , x i ] construct a rectangle that extends vertically from g ( x i * ) to f ( x i * ). Figure 6.3 (a) shows the rectangles when x i * is selected to be the left endpoint of the interval and n =10. Figure 6.3 (b) shows a representative rectangle in detail. Use this calculator (http://www.openstax.org/l/20_CurveCalc) to learn more about the areas between two curves.
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