Meaning of integral | Babel Free

Definitions

1. Abbreviation of International Gamma-Ray Astrophysics Laboratory (a satellite of the European Space Agency).
   abbreviation, alt-of
2. 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.
3. whole
4. Any of several analytic formalizations of this operation: the Riemann integral, the Lebesgue integral, etc.
   specifically
5. brown
6. 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.
7. wholegrain
8. 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;
9. The fluent of a given fluxion in Newtonian calculus.
   historical, obsolete

Equivalents

Examples

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

“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.”

“Stokes' theorem relates the integral over a surface of the curl of a vector field to a line integral around the boundary of that surface: #92;int#95;S#92;nabla#92;times#92;mathbf#123;F#125;#92;cdotd#92;mathbf#123;S#125;#61;#92;int#95;#123;#92;partialS#125;#92;mathbf#123;F#125;#92;cdotd#92;mathbf#123;r#125;,.”

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

CEFR level