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lecture32a

# lecture32a - Lecture 32(Chapter 5 Sec 32 The Denite...

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Lecture 32 (Chapter 5, Sec. 32): The Definite Integral Def. If f is defined for a x b , divide [ a, b ] into n subintervals of equal width Δ x = Let x 0 (= a ) , x 1 , x 2 , ..., x n (= b ) be the endpoints of these subintervals and let x * i be any sample point in the subinterval [ x i - 1 , x i ]. The definite integral of f from a to b is if the limit exists. If so, f is integrable on [ a, b ]. The sum n X i =1 f ( x * i x is a It is used to approximate the definite integral.

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2 Notation Integral sign Integrand Integration Limits of integration (lower and upper) dx NOTE: Theorem: If f is continuous or has a finite number of jump discontinuities on [ a, b ], then f is integrable on [ a, b ].
3 ex. Express lim n →∞ n X i =1 x * i e ( x * i ) 2 - 3 Δ x as a definite integral on [0 , 4]. Riemann Sums, Definite Integral, and Area: If f ( x ) 0 on [ a, b ] 6 - ? If f ( x ) 0 for some x in [ a, b ] 6 - ?

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4 How to evaluate definite integrals? Evaluating Definite Integrals as Area ex. Evaluate Z 6 - 2 | 4 - x | dx .
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lecture32a - Lecture 32(Chapter 5 Sec 32 The Denite...

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