3 - Integration

Question 1

Define a subdivision P of [a,b]

Answer 1

P is a set satisfying
1. P ⊂ [a,b]
2. a, b ∈ P

Question 2

Define Lower Riemann sum L(f,P)

Answer 2

For P = {x_i, 0 ≤ i ≤ n}
L(f,P) = (i=1 to n)∑ (x_i - x_i-1) inf f[x_i-1, x_i]

Question 3

Define Upper Riemann sum U(f,P)

Answer 3

For P = {x_i, 0 ≤ i ≤ n}
U(f,P) = (i=1 to n)∑ (x_i - x_i-1) sup f[x_i-1, x_i]

Question 4

Define Lower Riemann integral.

Answer 4

(a- to b) ∫ f = sup{ L(f, P) | P is a subdivision of [a,b] }

Question 5

Define Upper Riemann integral.

Answer 5

(a to b-) ∫ f = inf{ U(f, P) | P is a subdivision of [a,b] }

Question 6

Define what it means for a function to be (Riemann) integrable.

Answer 6

(a- to b) ∫ f = (a to b-) ∫ f
(f: [a,b] → R bounded)

Question 7

State the Cauchy criterion for integrability.

Answer 7

Let f: [a, b]→R be a
bounded function. Then f is integrable if and only if :
∀ε > 0 ∃P subdivision of [a, b] such that U(f, P) − L(f, P) < ε

Question 8

What’s the conclusion if f: [a, b]→R is monotone increasing or decreasing?

Answer 8

f is integrable on [a, b]

Question 9

What’s the conclusion if f: [a, b]→R is continuous on [a,b]?

Answer 9

f is integrable on [a, b]

Question 10

Define uniformly continuous

Answer 10

Let I ⊂ R be an interval. If function f : I → R is such that
∀ε>0. ∃δ>0 such that ∀x, y ∈ I |x−y| < δ ⇒ |f(x) − f(y)| < ε,
then f is uniformly continuous.

Question 11

What can be concluded if f: [a, b]→R is continuous on [a, b]?

Answer 11

f is uniformly continuous on [a, b].

Question 12

Explain additivity of the integral

Answer 12

Let f: [a, b]→R be bounded. For any a < c < b, f is integrable in [a, b] iff f is integrable in [a, c] and f is
integrable in [c, b]. Then,
{a to c}∫ f + {c to b}∫ f = {a to b}∫ f.

Question 13

Explain linearity of the integral.

Answer 13

For integrable functions f,g: [a, b] R and α,β ∈ R, αf + βg is integrable in [a, b], and
{a to b}∫ (αf + βg) = α* {a to b}∫ f + β* {a to b}∫ g

Question 14

First Fundamental Theorem of Calculus

Answer 14

Let F: [a, b]→R be continuous on [a, b] and differentiable on (a, b). Let f: [a, b]→R be integrable such that f(x) = F′(x) for x ∈ (a, b). Then:
{a to b}∫ f = F(b) − F(a)

Question 15

Second Fundamental Theorem of Calculus

Answer 15

Suppose f: [a, b]→R is integrable on [a,b]. Define F: [a, b]→R to be F(x) = {a to x}∫ f. Then:
1. F is continuous on [a, b]
2. If f is continuous at c ∈ [a, b], f is differentiable at c and F’(c) = f(c)

Question 16

Define what it means for F to be a primitive of f

Answer 16

f: [a, b]→R is such that f(x) = F′(x) for x ∈ (a, b), where F: [a, b] → R is continuous on [a, b] and differentiable on (a, b).

Question 17

Integration by parts

Answer 17

Let f, g: [a, b]→R be continuous and F, G: [a, b]→R be primitives of f and g respectively. Then
{a to b}∫ fG + {a to b}∫ Fg = F(b)G(b) − F(a)G(a).

Question 18

Integration by substitution

Answer 18

Let J ⊂ R be an open interval
and u: J→R be continuously differentiable. Let I ⊂ R be a closed interval and
f: I→R be continuous on I. Assume that u(x) ∈ I for all x ∈ J. Then for any a, b ∈ J
{u(a) to u(b)}∫ f = {a to b}∫ (f◦u)*u′