notes/docs/mathematics/calculus/integration.md

Integration

Sigma notation

if m and n are integers with m \leq n, and if f is a function defined as f: \{m,m+1,...,n\} \to \mathbb{R}, the symbol \sum_{i=m}^{n} f(i) represents the sum of the values of f at those integers:

\sum_{i=m}^{n} f(i) = f(m) + f(m+1) + f(m+2) + ... + f(n).

The explicit sum appearing on the right side of this equation is the expansion of the sum represented in sigma notation on the left side.

Partitions

Let P be a finite set of points arranged in order between a and b on the real line

P = {x_0, x_1, ... , x_{n-1}, x_n},

where a = x_0 < x_1 < ... < x_{n-1} < x_n = b. Such a set P is called a partition of [a,b]; it divides [a,b] into n subintervals of which the ith is $[x_{i-1},x_i]. The length of the ith subinterval of P is

\Delta x_i = x_i - x_{i-1} \quad \mathrm{for} \space 1 \leq i \leq n,

Then, the norm of the partition P is defined as

\parallel P \parallel = \max_{1 \leq i \leq n} \Delta x_i

If the function f is continuous on the interval [a,b], it is continuous on each subinterval [x_{i-1},x_i], and has a maximum u_i and minimum l_i on each subinterval by the Extreme value theorem such that

f(l_i) \leq f(x) \leq f(u_i) \quad \forall x \in [x_{i-1},x_i]/

Upper and lower Riemann sums

The lower Riemann sum, L(f,P), and the upper Riemann sum, U(f,P), for the function f amd the partition P are defined by:

\begin{array}{ll} L(f,P) &= f(l_1)\Delta x_1 + f(l_2)\Delta x_2 + ... + f(l_n)\Delta x_n \ &= \sum_{i=1}^n f(l_i)\Delta x_i, \ U(f,P) &= f(u_1)\Delta x_1 + f(u_2)\Delta x_2 + ... + f(u_n)\Delta x_n \ &= \sum_{i=1}^n f(u_i)\Delta x_i. \end{array}

Theorem: for any partitions P, Q on [a,b] all lower Riemann sums are smaller than or equal to any upper Riemann sums:

L(f,P) \leq U(f,Q).

Proof: let P, Q be partitions on [a,b], suppose L(f,P) \leq U(f,Q), define R = P \cup Q, R is a refinement of P, Q. Then,

L(f,P) \leq L(f,R) \leq U(f,R) \leq U(f,Q).

The definite integral

Suppose there exists exactly one number I \in \mathbb{R} such that for every partition P of [a,b]:

L(f,P) \leq I \leq U(f,P).

Then the function f is integrable on [a,b] and I is called the definite integral

I = \int_a^b f(x) dx.

Theorem: suppose that a function f is bounded on the interval [a,b], then f is integrable on [a,b] if and only if \forall \varepsilon > 0 there exists a partition P of [a,b] such that

U(f,P) - L(f,P) < \varepsilon.

Proof: let a,b \in \mathbb{R}, \forall \varepsilon > 0 there is |a-b| < \varepsilon then a=b.

Theorem: if f is continuous on the interval [a,b], then f is integrable on [a,b].

Proof: is missing...

Properties

• If a \leq b and f(x) \leq g(x) \space \forall x \in [a,b]:

\int_a^b f(x)dx \leq \int_a^b g(x)dx.

Proof: is missing...

• The triangle inequality for sums extends to definite integrals. If a \leq b, then

|\int_a^b f(x)dx| \leq \int_a^b |f(x)|dx.

Proof: is missing...

• Integral of an odd function f(-x) = -f(x):

\int_{-a}^a f(x)dx = 0.

Proof: is missing...

• Integral of an even function f(-x) = f(x):

\int_{-a}^a f(x)dx = 2\int_0^a f(x)dx.

Proof: is missing...

The Mean-value theorem for integrals

If the function f is continuous on [a,b] then there exists a point c in [a,b] such that

\int_a^b f(x)dx = (b-a)f(c)

Proof: \forall x \in [a,b],

let m \leq f(x) \leq M,

m(b-a)=\int_a^b mdx \leq \int_a^b f(x)dx \leq \int_a^b Mdx = M(b-a),

m \leq \frac{1}{b-a} \int_a^b f(x)dx \leq M,

According to the intermediate value theorem there exists a c \in [a,b] such that

\frac{1}{b-a} \int_a^b f(x)dx = f(c)

Piecewise continuous functions

Let c_0 < c_1 < ... < c_n be a finite set of points on the real line. A function f defined on [c_0,c_n] except possibly at some of the points c_i, (0 \leq i \leq n), is called piecewise continuous on that interval if for each i (1 \leq i \leq b) there exists a function F_i continuous on the closed interval [c_{i-1},c_i] such that

f(x) = F_i(x).

In this case, te integral of f from c_0 to c_n is defined to be

\int_{c_0}^{c_n} f(x)dx = \sum_{i=1}^n \int_{c_i-1}^{c_i} F_i(x)dx.

The fundamental theorem of calculus

Suppose that the function f is continuous on an interval I containing the point a.

Part I. Let the function F be defined on I by

F(x) = \int_a^x f(t)dt.

Then F is differentiable on I, and F'(x) = f(x) there. Thus, F is an antiderivative of f on I:

\frac{d}{dx} \int_a^x f(t)dt = f(x).

Part II. If G(x) is any antiderivative of f(x) on I, so that G'(x) = f(x) on I, then for any b in I there is

\int_a^b f(x)dx = G(b) - G(a).

Proof: using the definitions of the derivative

\begin{array}{ll} F'(x) &= \lim_{h \to 0} \frac{F(x+h)-F(x)}{h}, \ &= \lim_{h \to 0} \frac{1}{h}(\int_a^{x+h} f(t)dt - \int_a^x f(t)dt), \ &= \lim_{h \to 0} \frac{1}{h} \int_x^{x+h} f(t)dt, \ &= \lim_{h \to 0} hf(c) \quad \mathrm{for \space some} \space c \in [x,x+h], \ &= \lim_{c \to x} f(c), \ &= f(x). \end{array}