Chapter 6 | Applications of Integration
625
Figure 6.3 (a)We can approximate the area between the graphs of two functions, f ( x ) and g ( x ), with rectangles. (b) The area of a typical rectangle goes from one curve to the other.
The height of each individual rectangle is f ( x i * )− g ( x i * ) and the width of each rectangle is Δ x . Adding the areas of all the rectangles, we see that the area between the curves is approximated by A ≈ ∑ i =1 n ⎡ ⎣ f ( x i * )− g ( x i * ) ⎤ ⎦ Δ x . This is a Riemann sum, so we take the limit as n →∞ and we get A = lim n →∞ ∑ i =1 n ⎡ ⎣ f ( x i * )− g ( x i * ) ⎤ ⎦ Δ x = ∫ a b ⎡ ⎣ f ( x )− g ( x ) ⎤ ⎦ dx . These findings are summarized in the following theorem. ⎦ . Let R denote the region bounded above by the graph of f ( x ), below by the graph of g ( x ), and on the left and right by the lines x = a and x = b , respectively. Then, the area of R is given by (6.1) A = ∫ Theorem 6.1: Finding the Area between Two Curves Let f ( x ) and g ( x ) be continuous functions such that f ( x ) ≥ g ( x ) over an interval ⎡ ⎣ a , b ⎤ ⎡ ⎣ f ( x )− g ( x ) ⎤ ⎦ dx .
a b
We apply this theorem in the following example.
Example 6.1 Finding the Area of a Region between Two Curves 1
If R is the region bounded above by the graph of the function f ( x ) = x +4 and below by the graph of the function g ( x ) =3− x 2 over the interval [1, 4], find the area of region R .
Solution The region is depicted in the following figure.
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