integral

Pluralintegrals

1. 1
   adjective

   Forming an essential part of a whole, so that the thing is incomplete without it.

   Volunteers were integral to the success of the entire festival.

   Her research played an integral role in the company's new design.

   Similaressential, necessary, intrinsic, inherent, ingrained

   Oppositeperipheral, incidental, inessential, extraneous

   Note

   The everyday sense, and the one that stays closest to its Latin source, integer, "entire": a part is integral when the whole is not entire without it. Robert South already pairs the word with parts: "the two great integral parts that complete this duty." Inherent and necessary come close, but integral stresses belonging to a whole rather than simply being required. It often takes to, as in integral to the plan.

2. 2
   nouncountable

   One of the two basic operations of calculus, which finds quantities such as area, volume, or displacement by summing infinitely many infinitesimal parts.

   She used an integral to find the total distance the car traveled.

   The integral sums countless thin slices to yield the area beneath the graph.

   Oppositederivative

   Note

   Calculus has two halves, and this is the accumulating one, paired with differentiation. It is built as a limit of partial sums and written with ∫, an elongated long s. The word brings its own adjective for functions: one that has an integral is integrable. The caution matters, because not every function is. In ordinary use you take, find or compute the integral of a function.

3. 3
   nouncountable

   The numerical result of integrating a function over a given range, such as an area or total: a value that may be finite or infinite.

   The integral over that interval converges to a finite value.

   Because the area was unbounded, the integral diverged to infinity.

   Note

   The number-valued integral, the one with limits. Its verbs are converge and diverge: the integral of 1/x from ½ to 1 is ln 2, but over (0,1] it diverges. It takes over or on for the domain, as in the integral over a surface, and compounds can name the domain, as a line integral runs along a curve. Interval endpoints sit as sub- and superscripts on ∫.

4. 4
   adjective

   Relating to or involving integration in calculus.

   Integral calculus provides tools for computing areas and volumes.

   The physicist took an integral approach to model the accumulated charge.

   Oppositedifferential

   Note

   The adjective of calculus, met mostly in fixed phrases such as integral calculus and integral sign, where it means "belonging to integration". Its natural partner is differential, the adjective of the other half of calculus. Outside such technical phrases the word drifts back toward the "essential part" meaning, so a phrase like an integral approach is more naturally read as "essential" than as "by integration".

5. 5
   adjective

   Relating to or consisting of integers; whole-numbered rather than fractional.

   The equation has only integral solutions, such as 2 and 5.

   We restricted the variable to integral values.

   Oppositefractional

   Note

   Here the Latin integer survives as the mathematical noun itself, so the adjective means "of the integers": integral values, integral solutions. Context has to keep it apart from the calculus adjective. Integral values are whole numbers, while integral calculus has nothing to do with whole numbers. When the two could collide, the surrounding vocabulary of equations or of areas usually settles which one is meant.

6. 6
   nouncountable

   The result of integrating a function without fixed limits, producing another function; the function's antiderivative.

   Every integral of this kind includes an arbitrary constant.

   She computed the integral to recover the original function from its derivative.

   Similarantiderivative

   Oppositederivative

   Note

   Here the answer is a function, not a number. The integral of x² is x³/3 plus a constant, and that constant is never optional, because the result is fixed only up to it. Antiderivative names the same thing more plainly, and derivative is its opposite. Unlike the definite integral, it carries no limits on the sign.

7. 7
   nouncountable

   Any of several rigorous mathematical definitions of integration, such as the Riemann integral or the Lebesgue integral.

   The Lebesgue integral extends the older Riemann integral to a much broader class of functions.

   Mathematicians choose which integral to use depending on the functions involved.

   Note

   The countable sense. Once integration was made rigorous, it was made rigorous more than once, so here the word takes an article and a name: the Riemann integral, the Lebesgue integral. Each is a separate formal account of the calculus operation, built on differing assumptions. A careful claim that a function is integrable therefore carries an unspoken "in which sense?"

8. 8
   adjective

   In commutative algebra, said of a ring element relative to a subring A: being a root of some monic polynomial with coefficients in A.

   Every element of the smaller ring is integral over the base ring.

   An algebraic integer is integral over the integers.

   Note

   A relative property, always stated with over: an element is integral over a subring, never simply integral. The monic condition, a leading coefficient of 1, is what gives it bite. Over a field every algebraic element qualifies, but over the integers that is not so. The name echoes the "integer" meaning. Complex numbers integral over the integers are exactly the algebraic integers, and the elements integral over A form a ring, its integral closure.

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