Definition of Definite Integrals

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Definition of a Riemann Sum
Let f be defined on the closed interval [a, b], and let ¤ be a partition of [a, b] given by a = x₀ < x₁ < x₂ < ... < x_n-1 < x_n = b, where ¤x₁ is the length of the ithe subinterval. If c_i is any point in the ithe subinterval, then the sum
 _n
£   f(c_i) ¤x_i,   x_{i - 1} < c_i < x_i
^{i = 1}
is called a Riemann sum of f for the partition ¤.

Definition of a Definite Integral
If f is defined on the closed interval [a, b] and the limit
             _n
      lim  £  f(c_i) ¤x_i
  ^{||¤||-->0  i = 1}
exists, then f is integrable on [a, b] and the limit is denoted by
             _n
      lim  £  f(c_i) ¤x_i =
  ^{||¤||-->0  i = 1}
  _b
§  f(x) dx
  ^a
The limit is called the definite integral of f from a to b. The number a is the lower limit of the integration, and the number b is the upper limit of integration.

Continuity Implies Integrability
If a function f is on the closed interval [a, b], then f is integrable on [a, b].

Two Special Definite Integrals
If f is defined at x = a, then
  _a
§  f(x) dx = 0.
  ^b

If f is integrable on [a, b], then
  _a
§  f(x) dx =
  ^b
    _b
-§  f(x) dx.
    ^a