4.2 The Definite Integral

4.2 The Definite Integral - closed interval [ a, b ], then...

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The Definite Integral Section 4.2
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Defn. of a Riemann Sum Let f be defined on the closed interval [a, b], and let be a partition of [a, b] given by a = x0 < x1 < x2 < . . .< xn-1 < xn= b, Where is the length of the i th subinterval. If ci is any point in the i th subinterval, then the sum i x
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xi-1< ci < xi is called a Riemann sum of f for the partition . ci = a + i 1 ( ) n i i i f c x = x
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The width of the largest subinterval of a partition is the norm of the partition and is denoted by . If every subinterval is of equal length, the partition is regular and the norm is denoted by b a x n - ∆ = ∆ =
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Defn. of a Definite Integral If f is defined on the closed interval [a, b] and the limit exists, then f is integrable on [a, b], and the limit is denoted by 0 1 lim ( ) n i i i f c x ∆ → = 0 1 lim ( ) ( ) b n i i i a f c x f x dx ∆ → = =
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Continuity Implies Integrability If a function f is continuous on the closed interval [a, b], then f is integrable on [a, b].
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Thm. 4.5 The Definite Integral as the Area of a Region If f is continuous and nonnegative on the
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Unformatted text preview: closed interval [ a, b ], then the area of the region bounded by the graph of f , the x-axis, and the vertical lines x = a and x = b is given by f (x)dx a b Defn. of 2 Special Definite Integrals 1. If f is defined at x = a, then 2. If f is integrable on [a, b], then f (x)dx a a = 0 f (x)dx a = - f (x)dx a b Thm. 4.6 Additive Interval Property If f is integrable on the three closed intervals determined by a, b, and c, then f (x)dx a b = f (x)dx a c + f (x)dx c b Properties of Definite Integrals Preservation of Inequality 1. If f is integrable and nonnegative on the closed interval [ a, b ], then 2. If f and g are integrable on the closed interval [ a, b ] and f(x) &lt; g(x) for every x in [ a, b ], then f (x)dx a b 0 &lt; f (x)dx a b &lt; g (x)dx a b Joke Time What do you call 2 spiders who just got married? Newlywebs What did the Atlantic Ocean say to the Pacific Ocean? Nothing, they just waved....
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This note was uploaded on 12/01/2011 for the course MATH 112 taught by Professor Jarvis during the Fall '08 term at BYU.

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4.2 The Definite Integral - closed interval [ a, b ], then...

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