Integration

Question 1

defining integration for step functions

Answer 1

I(f) = signed area of the rectangles
= ∑ (xk+1 - xk)f(xk)
where xk is any element in (xk,xk+1)

Question 2

defining integration for regulated functions

Answer 2

if fn-> f regulated
define I(f) = lim I(fn) as n->∞
(independent of choice of step function)

Question 3

equidistant partition

Answer 3

xk = a + k(b-a)/n
for x from a to b with n partitions
set fn(x) = f(xk) for x in [xk,xk+1) or f(xn) for x = xn =b

Question 4

properties of integration

Answer 4

linearity
- ∫c.f(x)dx = c ∫f(x) dx
- ∫(f+g) dx = ∫ f dx + ∫ g dx
additivity
- ∫ from c to a + ∫ from b to c = ∫ from b to a
monoticity
- f(x) ≤ g(x) for all x in [a,b]
then ∫f(x) dx ≤ ∫g(x) dx
- if f(x) ≥ 0 then ∫ f(x) ≥ 0
- if m = inf( f(x)) and M = sup( f(x)) then m ≤f(x) ≤M and m(b-a)≤ ∫f(x) ≤ M(b-a)

Question 5

mean value theorem for integrals

Answer 5

f CONTINUOUS on [a,b]
then there exists a c in [a,b] st ∫ f(x) from b to a = f(c).(b-a)

Question 6

if f(x) ≥ 0 and a point >0

Answer 6

f(x) cont on [a,b] assume f(x) ≥ 0 and f(c) >0 fro some c in [a,b] then ∫ f(x) dx >0

Question 7

if regulated what is inetgral

Answer 7

if f regulated on [a,b] then F(x) is cont on [a,b] in fact lipschitz
(where F(x) = ∫ f(t) dt from x to a)

Question 8

FTC

Answer 8

f [a,b] ->R cont then
F(x) = ∫ f(t) dt from x to a is diff on [a,b] and F’(x) = f(x)
in particular every continuous function has an anti-derivative

Question 9

FTC if f only regulated not cont

Answer 9

can show f still has some kind of anti derivative which is left and right differentiable but not necessarily the same

Question 10

functions by integrals

Answer 10

can define functions as an integral eg
L(x) = ∫ 1/t dt from x to 1 would be log(x)
then can define properties of the function from properties of the inetgral