![]() Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. We recommend using aĪuthors: Gilbert Strang, Edwin “Jed” Herman Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses theĬreative Commons Attribution-NonCommercial-ShareAlike License The expression x()2 + y()2 can be simplified a great deal we leave this as an exercise and state that x()2 + y()2 f()2 + f()2. We want to find the area between the graphs of the functions, as shown in the following figure. We compute x() and y() as done before when computing dy dx, then apply Equation 9.5.17. Let u ( y ) u ( y ) and v ( y ) v ( y ) be continuous functions over an interval such that u ( y ) ≥ v ( y ) u ( y ) ≥ v ( y ) for all y ∈. Let’s develop a formula for this type of integration. Therefore, if we integrate with respect to y, y, we need to evaluate one integral only. When the graphs are represented as functions of y, y, we see the region is bounded on the left by the graph of one function and on the right by the graph of the other function. However, based on the graph, it is clear we are interested in the positive square root.) Similarly, the right graph is represented by the function y = g ( x ) = 2 − x, y = g ( x ) = 2 − x, but could just as easily be represented by the function x = u ( y ) = 2 − y. (Note that x = − y x = − y is also a valid representation of the function y = f ( x ) = x 2 y = f ( x ) = x 2 as a function of y. We could just as easily solve this for x x and represent the curve by the function x = v ( y ) = y. ![]() Note that the left graph, shown in red, is represented by the function y = f ( x ) = x 2. What if we treat the curves as functions of y, y, instead of as functions of x ? x ? Review Figure 6.7. However, there is another approach that requires only one integral. In Example 6.4, we had to evaluate two separate integrals to calculate the area of the region. We want to find the area between the graphs of the functions, as shown in the following figure.Ĭonsider the region depicted in the following figure. Let f ( x ) f ( x ) and g ( x ) g ( x ) be continuous functions over an interval such that f ( x ) ≥ g ( x ) f ( x ) ≥ g ( x ) on. Last, we consider how to calculate the area between two curves that are functions of y. We then look at cases when the graphs of the functions cross. ![]() We start by finding the area between two curves that are functions of x, x, beginning with the simple case in which one function value is always greater than the other. In this section, we expand that idea to calculate the area of more complex regions. In Introduction to Integration, we developed the concept of the definite integral to calculate the area below a curve on a given interval. 6.1.3 Determine the area of a region between two curves by integrating with respect to the dependent variable.6.1.2 Find the area of a compound region.6.1.1 Determine the area of a region between two curves by integrating with respect to the independent variable.
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