This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ...
View
Full
Document
 Fall '08
 ALL
 Calculus, Limits, dx, Riemann, dx c dx, integration dx Note

Click to edit the document details