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Line 14: − Integration is the inverse function of [[differentiation]]; the two notions are related by the [[Fundamental Theorem of Calculus]]. Thus, if <math>\int f(x) dx = g(x)</math>, then <math>\frac{d}{dx}g(x) = f(x)</math>.+ + + + + + + + + +
Added definition. We need graphs! Could someone with image upload rights help?
==Integration==
==Integration==
===Definition===
An integral 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>.
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.
Indefinite integrals are related to definite integrals by the second part of the [[Fundamental Theorem of Calculus]].
===Applications===
The first part of the [[Fundamental Theorem of Calculus]] states that integration is the reverse function of [[derivative|differentiation]]. Thus, if <math>\int f(x) dx = g(x)</math>, then <math>\frac{d}{dx}g(x) = f(x)</math>.
Note that, because the derivative of any constant function is 0, <math>\frac{d}{dx}x^2+2 = \frac{d}{dx}x^2+3 = \frac{d}{dx}x^2 = 2x</math>. Therefore, <math>\int 2x dx</math> does not simply equal x<sup>2</sup>, but rather x<sup>2</sup> + C for any constant real number C. The above-mentioned functions are the ''family of antiderivatives'' for 2x.
Note that, because the derivative of any constant function is 0, <math>\frac{d}{dx}x^2+2 = \frac{d}{dx}x^2+3 = \frac{d}{dx}x^2 = 2x</math>. Therefore, <math>\int 2x dx</math> does not simply equal x<sup>2</sup>, but rather x<sup>2</sup> + C for any constant real number C. The above-mentioned functions are the ''family of antiderivatives'' for 2x.