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Unformatted text preview: Lecture 27 Indefinite Integrals, Net Change To evaluate a definite integral on [ a,b ] using FTCII, we: (1) (2) Evaluate the definite integral Z b a ( x 3 x ) d x when a = 1 and b = 2 . When a = 0 and b = 4 ... Regardless of a and b , step (1), remains the same, so a convenient notation for the general antiderivative of f ( x ) is: This is called an Note that a definite integral is a whereas an indefinite integral is a . Calculate the indefinite integrals: Z 1 + x x d x Z tan 2 ( ) d In Lecture 24, we explained how the area under the rate of change of f is the accumulated (net) change in the value of f on the interval. FTCII states this more generally even when f is negative: Z b a f ( x ) d x = f ( b ) f ( a )  {z }  {z } When stated this way, this is often called the If the volume V of water in a lake is changing at rate ( t ) , with respect to time, then Z b a ( t ) d t = If a population P is changing at rate ( t ) at time t , then Z b a ( t ) d t = If the force...
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 Spring '08
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 Calculus, Integrals

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