Lecture3bc3 - • find an antiderivative; • evaluate at...

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Instructor: Quanlei Fang Dept. of Math, University at Buffalo, Spring 2009 Welcome to Math 122 Lecture 3 Survey of Calculus and its Applications I1 ± Integral calculus MTH 121 Review MTH 121 Review
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Integration Integration Δ x = ( b a )/ n , x 1 , x 2 , …., x n are selected points from a partition [ a , b ]. Definite Integral Brief Brief Antiderivative Antiderivative Review Review
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Properties of Definite Integrals Properties of Definite Integrals Area and definite integral Area and definite integral f ( x ) a b dx = Area above Area below f ( x ) 0 4 dx =
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Area between two curves Area between two curves a b Top function Bottom function dx ( g ( x ) f ( x )) i 1 i 2 dx + ( f ( x ) g ( x )) i 2 i 3 dx Area between two curves Area between two curves
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Fundamental theorem of Calculus Fundamental theorem of Calculus Theorem: (Fundamental Theorem I) If F is a differentiable function on [a,b], then Or: If f is continuous on [a,b], F is an antiderivative for f , then To solve a definite integral, we need to
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Unformatted text preview: • find an antiderivative; • evaluate at b; • evaluate at a; • subtract the results. Example Example Given f (2) = 1, ′ f ( x ) 2 ∫ dx = 7, f (0) = ? x + x 4 ∫ dx = Average value Average value Average Value of f over the interval [a,b] = 1 b − a f ( x ) dx a b ∫ EXAMPLE EXAMPLE SOLUTION SOLUTION ( Average Temperature ) During a certain 12-hour period the temperature at time t (measured in hours from the start of the period) was degrees. What was the average temperature during that period? 2 3 1 4 47 t t − + Volume Volume EXAMPLE EXAMPLE SOLUTION SOLUTION Find the volume of a solid of revolution generated by revolving about the x-axis the region under the following curve. 2 to 1 from 2 = = = x x x y From next week From next week Chapter 7 Functions of several variables...
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This note was uploaded on 02/20/2009 for the course MTH 122 taught by Professor Buettgens during the Spring '08 term at SUNY Buffalo.

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Lecture3bc3 - • find an antiderivative; • evaluate at...

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