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calc notes lecture 29

calc notes lecture 29 - = 2 4 Example 3 Determine the area...

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Lecture Section Objectives Assignment 30 5.5 The Area of a region Bounded by Two Graphs 5.5: 1-7 odd, 15-29 odd, 35, 37 , 51 Understanding Goals: 1. Understand how to find the areas of regions bounded by two graphs. 2. Understand how to solve real-life problems using the areas of regions bounded by two graphs. 1
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The Area between Curves Case I : The area between two continuous functions ( 29 ( 29 x g y x f y = = and on the interval [a, b] with ( 29 ( 29 x g x f [ ] ( ) ( ) b a A f x g x dx = - or ( 29 ( 29 [ upper function lower function ] , b a A dx a x b = - Case II: The area between two continuous functions ( 29 ( 29 y g x y f x = = and on the interval [c, d] with ( 29 ( 29 y g y f ( 29 ( 29 [ ] d c A f y g y dy = - or ( 29 ( 29 [ right function left function ] , d c A dy c y d = - * Note that if g f the area between f and g is ( 29 ( 29 b a f x g x dx - , regardless of the signs of f and g . 2
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Example 1 : Find the area bounded by 2 , 2 1, 2, and the -axis. x y xe y x x y - = = + = 3
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Example 2 : Determine the area of the region enclosed by ( 29 ( 29 x x g x
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Unformatted text preview: = , 2 4 Example 3 : Determine the area of the region bounded by 5 and 2 , 16 4 , 10 2 2 =-= + = + = x x x y x y 5 Sometimes it is easier to integrate with respect to y than with respect to x to find the area between two curves. Example 4 : Determine the area of the region bounded by ( 29 2 2 2 and 10-= +-= y x y x 6 Example 5 : An epidemic was spreading such that t weeks after its outbreak it had infected 2 1 ( ) 0.1 0.5 150 N t t t = + + , 0 50 t g people. Twenty-five weeks after the outbreak, a vaccine was developed and administered to the public. At that point, the number of people infected was governed by the model 2 2 ( ) 0.2 6 200 N t t t = -+ + . Approximate the number of people that the vaccine prevented from becoming ill during the epidemic. 7...
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