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Unformatted text preview: Math 006 (Lecture 32) Deﬁnite Integrals
Deﬁnition 1. Suppose is
a f (x)dx = F (x), then the deﬁnite integral of f (x) from a to b
b f (x)dx = F (b) − F (a). The lower limit of integration is a and the upper limit of integration is b. Example 1. Find
2 4 (a)
0 x2 dx (b)
1 (ln x)2 dx x 1/2 (c)
−1 xex dx 2 Meaning of deﬁnite integral: The deﬁnite integral,
b f (x)dx,
a represents the cumulative sum of the signed areas between the graph of f and the xaxis from x = a to x = b Example 2. Find
1 0 (a)
0 xdx (b)
−1 xdx. 1 Example 3. Calculate the deﬁnite integrals with the indicated area:
b (a)
a f (x)dx c (b)
a f (x)dx c (c)
b f (x)dx Properties of deﬁnite integrals
a 1.
a b f (x)dx = 0
a 2.
a b f (x)dx = −
b f (x)dx
b 3.
a b kf (x)dx = k
a f (x)dx
b b 4.
a b [f (x) ± g (x)]dx =
a c f (x)dx ±
a b g (x)dx 5.
a f (x)dx =
a f (x)dx +
c f (x)dx Deﬁnite integrals and Substitution technique
Example 4. Evaluate: 5 x (a) dx 2 0 x + 10
3 (b)
2 √ x 2x2 − 3dx Deﬁnition 2. The average value of a continuous function f (x) on [a, b] is given by 1 b−a
b f (x)dx.
a Example 5. Given that the demand function p = D(x) = 100e−0.05x , ﬁnd the average price over the demand interval [40, 60]. 2 ...
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 Fall '09
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 Math, Integrals

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