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Chapter 6 | Applications of Integration
Areas of Compound Regions So far, we have required f ( x ) ≥ g ( x ) over the entire interval of interest, but what if we want to look at regions bounded by the graphs of functions that cross one another? In that case, we modify the process we just developed by using the absolute value function. Theorem 6.2: Finding the Area of a Region between Curves That Cross Let f ( x ) and g ( x ) be continuous functions over an interval ⎡ ⎣ a , b ⎤ ⎦ . Let R denote the region between the graphs of f ( x ) and g ( x ), and be bounded on the left and right by the lines x = a and x = b , respectively. Then, the area of R is given by A = ∫ a b | f ( x )− g ( x ) | dx . In practice, applying this theorem requires us to break up the interval ⎡ ⎣ a , b ⎤ ⎦ and evaluate several integrals, depending on which of the function values is greater over a given part of the interval. We study this process in the following example. Example 6.3 Finding the Area of a Region Bounded by Functions That Cross
If R is the region between the graphs of the functions f ( x ) = sin x and g ( x ) =cos x over the interval [0, π ], find the area of region R .
Solution The region is depicted in the following figure.
Figure 6.6 The region between two curves can be broken into two sub-regions.
The graphs of the functions intersect at x = π /4. For x ∈ [0, π /4], cos x ≥ sin x , so | f ( x )− g ( x ) | = | sin x −cos x | =cos x −sin x . On the other hand, for x ∈ [ π /4, π ], sin x ≥cos x , so
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