6 0. Background

Theorem 0.14. (Fundamental Theorem of Calculus, Second Form) Suppose that

f(x) is Riemann-integrable on the interval [a, b]. Suppose further that F (x) is a

differentiable function on the interval [a, b] such that F (x) = f(x) for a ≤ x ≤ b.

Then

b

a

f(x) dx = F (b) − F (a) .

The Second Form of the Fundamental Theorem follows from the First Form if

f is continuous. The point of the Second Form is that it holds under the weaker

assumption that f is Riemann-integrable.

Theorem 0.15. (Integration by parts) Suppose that f is Riemann-integrable on

[a, b], that F is a function such that F (x) = f(x), and that g is a differentiable

function such that g (x) is Riemann-integrable on [a, b]. Then

b

a

f(x)g(x) dx = F (b)g(b) − F (a)g(a) −

b

a

F (x)g (x) dx .

This follows immediately from the Second Form of the Fundamental Theorem,

in view of the differentiation formula (Fg) = F g + Fg = fg + Fg .

Theorem 0.16. (The triangle inequality for integrals) If f is Riemann-integrable,

then

b

a

f(x) dx ≤

b

a

|f(x)| dx .

Theorem 0.17. (Leibniz’s Rule) If f(x, y) and

∂

∂x

f(x, y) exist and are continuous

on the closed rectangle a ≤ x ≤ b, c ≤ y ≤ d, then the function

F (x) =

d

c

f(x, y) dy

is differentiable for a ≤ x ≤ b, and

F (x) =

d

c

∂

∂x

f(x, y) dy .

Theorem 0.18. Suppose that f(x) =

∑∞

n=1

fn(x). If the functions fn are differen-

tiable, and if the series

∑∞

n=1

fn(x) is uniformly convergent, then f is differentiable,

and

f (x) =

∞

n=1

fn(x) .

Theorem 0.19. (Dominated Convergence) If amn is a double sequence, and An

is such that limm→∞ amn = A exists, and if there is a sequence Mn such that

|amn| ≤ Mn for all m, and

∑∞n

n=1

Mn ∞, then

lim

m→∞

∞

n=1

amn =

∞

n=1

An .