CAB-Chp-5-4-NTs - F(b) plug in x=b to F(x) (notice this is...

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CalcAB Chp 5-4 Fundamental Theorem of Calculus NOTES 1. REM: INDEFINITE INTEGRAL -- f(x)dx = F(x)+C GENERAL solution: F(x) + C where C is unknown PARTICULAR solution: F(x) + # where C is known (solved for) DEFINITE INTEGRAL (D.I.) -- f(x) dx a b Answer to a definite integral is a number value . Must have a CONTINUOUS function f(x) on the interval [a, b] Important concept: Given f(x) is a rate, the definite integral is a quantity. An application we have been focusing on is area under the curve f(x) on the interval [a, b] . Three methods to evaluate the D.I.: (1) Get exact value of D.I. using special equations and geometry area formulas. (2) Get estimated value of D.I. using a Riemann sums. (3) Get exact value of D.I. using FUNDAMENTAL THEOREM OF CALCULUS (This lesson) . 2. NOW: FUNDAMENTAL THEOREM OF CALCULUS (FTC): = F(b) F(a) Notation interpretation --
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Unformatted text preview: F(b) plug in x=b to F(x) (notice this is anti-derivative) F(a) plug in x=a to F(x) NOTE: There is no +C involved for definite integrals; in particular when using FTC. This is because definite integral answers are not equations. 3. Ex(FTC)-- Evaluate each integral. (a) dx (b) dx (c) dx (d) dx (e) dx (f) dx <Answers> (a) = (b) D.I. not possible because f(x) is not continuous at x=0, have a bad interval (c) = 3 ln2 = ln8 (d) = = (e) = 4 (f) = 2e 3 + 1 4. Ex(FTC and integral properties) Find the area of the shaded region made by the equations y1 and y2 for their respective intervals. Given: y1 = and y2 = x 2 <Answer> Set up the integrals (notice the proper form of the notation): Evaluate the integrals: = and = Sum of the two D.I.'s: 2/3 + 7/3 = 3 The area of the shaded region. 1 1 2 4 x y2 y1...
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This note was uploaded on 02/21/2011 for the course MATH 408C taught by Professor Knopf during the Spring '10 term at University of Texas at Austin.

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CAB-Chp-5-4-NTs - F(b) plug in x=b to F(x) (notice this is...

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