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Unformatted text preview: L31 – The Deﬁnite Integral Def – If f is continuous on [ a, b ], divide [ a, b ] into n subintervals of equal width with x∗ any point in [ xi−1 , xi ], then the deﬁnite integral from a to b is i 1 Notation: integral sign integrand integration limits of integration dx Note: Riemann Sum as area: If f (x) ≥ 0 for x in [ a, b ] If f (x) ≤ 0 for some x in [ a, b ] 2 n Express lim n→∞ x∗ exi −3 ∆xi as a deﬁnite integral on [ 0, 5 ] . i
i=1 ∗ Evaluate 5 −3  5 − x  dx in terms of area. 3 Evaluate 4 −2 (3x − 2) dx in terms of area. What if we don’t have a common geometric shape? 4 If c is any constant and if n is a positive integer, then
n 1.
i=1 cai = n 2.
i=1 (ai + bi ) = n 3.
i=1 (ai − bi ) = n 4.
i=1 c= n 5.
i=1 i= n 6.
i=1 i2 = n 7.
i=1 i3 = 5 Evaluate 3 0 (x2 − 3x) dx . 6 The Midpoint Rule –
b a f (x) dx ≈ where ∆x = and xi = Use the Midpoint Rule with n = 3 to approximate
4 1 1 dx . 4x − 2 7 Properties of integrals:
a b a a f (x) dx f (x) dx c dx [f (x) ± g (x)] dx cf (x) dx f (x) dx b a b a b a b a 8 If f (x) = ﬁnd
2 −3 √2 4 − x2 x<0 x≥0 f (x) dx . 9 Comparison Properties of Integrals If f (x) ≥ 0 for a ≤ x ≤ b, then If f (x) ≥ g (x) for a ≤ x ≤ b, then If m ≤ f (x) ≤ M for a ≤ x ≤ b, then 10 π Show that ≤ 6 π 2 π 6 sin(x) dx ≤ π . 3 11 ...
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 Fall '08
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 Calculus, Limits

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