Changes

==Integration==

==Integration==

===Definition===

===Definition===

An integral is defined as the limit of a [[Riemann sum]]:

If <math> f</math> is a function of [[bounded variation]], then the [[Riemann integral]] of <math> f</math> is defined as the limit of a [[Riemann sum]]:

:<math>\int\limits_{x_i}^{x_N} f(x) dx = \lim_{dx\to 0} \sum_{i=1}^N f(x_i)*dx</math>.

:<math>\int\limits_{x_i}^{x_N} f(x) dx = \lim_{dx\to 0} \sum_{i=1}^N f(x_i)*dx</math>.

Therefore, <math>\int\limits_a^b f(x) dx</math> is the area under the curve f(x) between the vertical line x=a and the vertical line x=b.  <tt>a</tt> and <tt>b</tt> are called the ''limits'' of the integral, and <tt>f(x)</tt> is called the ''integrand''.  This type of integral is referred to as a ''definite integral'', or an integral between definite limits.

The fact that <math> f </math> is of bounded variation implies that the Riemann sum above converges and is independent of the choices of <math>x_1,\dots,x_N</math>.  For integration of measurable functions which may not have bounded variation, the [[Lebesgue integral]] must be used. It is easy to see that <math>\int\limits_a^b f(x) dx</math> is the area under the curve f(x) between the vertical line x=a and the vertical line x=b.  <tt>a</tt> and <tt>b</tt> are called the ''limits'' of the integral, and <tt>f(x)</tt> is called the ''integrand''.  This type of integral is referred to as a ''definite integral'', or an integral between definite limits.

Indefinite integrals are related to definite integrals by the second part of the [[Fundamental Theorem of Calculus]].

Indefinite integrals are related to definite integrals by the second part of the [[Fundamental Theorem of Calculus]].