# L31 - L31 – The Deﬁnite Integral Def – If f is...

This preview shows pages 1–11. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the 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

{[ snackBarMessage ]}

### Page1 / 11

L31 - L31 – The Deﬁnite Integral Def – If f is...

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online