integral

Meanings

1. 1
   noun

   the result of a mathematical integration

   F(x) is the integral of f(x) if dF/dx = f(x)

2. 2
   adj

   Constituting a whole together with other parts or factors; not omittable or removable.

   Ceasing to do evil, and doing good, are the two great integral parts that complete this duty.

3. 3
   adj

   Whole; undamaged.

   A local motion keepeth bodies integral.

4. 4
   noun

   One of the two fundamental operations of calculus (the other being differentiation), whereby a function's displacement, area, volume, or other qualities arising from the study of infinitesimal change are quantified, usually defined as a limiting process on a sequence of partial sums. Denoted using a long s: ∫, or a variant thereof.

   The integral of a univariate real-valued function is the area under its curve; but be warned! Not all functions are integrable!

5. 5
   noun

   A definite integral: the result of the application of such an operation onto a function and a suitable subset of the function's domain: either a number or positive or negative infinity. In the former case, the integral is said to be finite or to converge; in the latter, the integral is said to diverge. In notation, the domain of integration is indicated either below the sign, or, if it is an interval, with its endpoints as sub- and super-scripts, and the function being integrated forming part of the integrand (or, generally, differential form) appearing in front of the integral sign.

   The integral of #92;frac#123;1#125;#123;x#125; on #91;#92;frac#123;1#125;#123;2#125;,1#93; is #92;ln(2), but the integral of the same function on (0,1#93; diverges. In notation, #92;int#95;#92;frac#123;1#125;#123;2#125;¹#92;frac#123;1#125;#123;x#125;dx#61;#92;ln(2), but #92;int#95;0¹#92;frac#123;1#125;#123;x#125;dx#61;#92;infty.

6. 6
   noun

   An indefinite integral: the result of the application of such an operation onto a function together with an indefinite domain, yielding a function; a function's antiderivative;

   The integral of x² is #92;frac#123;x³#125;#123;3#125; plus a constant.

7. 7
   adj

   constituting the undiminished entirety

8. 8
   adj

   existing as an essential constituent or characteristic

Etymology

Borrowed from Middle French integral, from Medieval Latin integrālis, from Latin integer (“entire”); see integer.