5_4 - Section 5.4 Indenite Integrals and the Net Change...

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Section 5.4: Indefinite Integrals and the Net Change Theorem Instructor: Ms. Hoa Nguyen Reminder about the Fundamental Theorem of Calculus 1. If g ( x ) = x a f ( t ) dt , then g ( x ) = d dx x a f ( t ) dt = f ( x ). 2. b a f ( x ) dx = F ( b ) - F ( a ) where F is any antiderivative of f , i.e., F = f or F = f ( x ) dx (indefinite integral). The difference between indefinite and definite integrals A definite integral b a f ( x ) dx is a number . An indefinite integral f ( x ) dx is a function (or family of functions). The second part of the Fundamental Theorem of Calculus shows the connection between these two integrals: b a f ( x ) dx = F ( b ) - F ( a ) = F ( x )] b a = f ( x ) dx ] b a Table of Indefinite Integrals (check the hand-out) Example 1 of Section 5.4 (textbook): Example 2 of Section 5.4 (textbook): Example 3 of Section 5.4 (textbook): Example 4 of Section 5.4 (textbook): Example 5 of Section 5.4 (textbook): Applications The Net Change Theorem The integral of a rate of change is the net change: b a F ( x ) dx = F ( b ) - F ( a ). Example : If V ( t ) is the volume of water in a reservoir at time t , its derivative V ( t ) is the rate at which water flows into the reservoir at time t . So t 2 t 1 V (
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