Unit Five Notes--Integration and Accumulation of Change

Unit Five Notes--Integration and Accumulation of Change -...

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Unformatted text preview: Unit Five Brief Calculus 5.1/5.2 Results of Change and Area Approximations Area Beneath A Curve Let f be a continuous, on an interval [a, b], non-negative bounded function from a to b. The area of the region between the graph of f and the x-axis from a to be is given by the following limit: Area = ( 29 ( 29 [ ] x x f x f x f n n + + + - ) ( . . . lim 2 1 , where x 1 , x 2 , . . . x n are the midpoints of n subintervals of length n a b x- = . Accumulated Change and the Definite Integral The accumulated change in f from a to b is given by the following limit: ( 29 x c f n i i = 1 || || lim = ( 29 ( 29 [ ] x x f x f x f n n + + + - ) ( . . . lim 2 1 = ( 29 dx x f b a , where x 1 , x 2 , . . . x n are the midpoints of n subintervals of length n a b x- = . The statement ( 29 dx x f b a is called the definite integral of f from a to b. Ex 1: Find the upper and lower sums for the region bounded by the graph of f(x) = x 2 and the x-axis and on [0, 2]. Use n = 4 subintervals. Find two approximations (using left endpoint and right endpoint approximations) of the area of the region lying between the graph of f and the x-axis. Ex 2) f(x) = -x 2 + 6 and [0, 2]. Use n = 4 subintervals. Find two approximations (using left endpoint and right endpoint approximations) of the area of the region lying between the graph of f and the x-axis. Ex 3. The rate of change of the per capita consumption of wine in the United States from 1970 through 1990 can be modeled as W(x) = 977 . 71 6174 . 2 0314 . 0001243 . 2 3- +- x x x gallons per person per year, where x is the number of years since the end of 1900, x > 0. A) Find the input value (zero) of W(x) at point A. B) From 1970 through 1990, according to the model, when was wine consumption increasing and when was it decreasing? C) Use a limiting value of sums to estimate the area or R 1 to two decimal places. D) According to the model, what was the change in the per capita consumption of wine from the end of 1970 to 1990? 5.3 Notes (Antiderivative) Definite and Indefinite Integral Now we have been introduced to 3 very important parts of Calculus:...
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Unit Five Notes--Integration and Accumulation of Change -...

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