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# lecture32 - Math 006(Lecture 32 Deﬁnite Integrals...

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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 x-axis 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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lecture32 - Math 006(Lecture 32 Deﬁnite Integrals...

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