Where does integral collide?

In calculus, the integral is the operation that adds a quantity up across a continuous range, together with the number or function that operation yields: the definite integral of a function measures the signed area between its graph and the axis, while the indefinite integral, or antiderivative, is the family of functions whose rate of change returns the original. Building on Newton and Leibniz, whose fundamental theorem of calculus ties the two together, and on Riemann's limit of ever finer rectangular sums, Lebesgue's 1902 Intégrale, longueur, aire rebuilt the integral around measuring sets of points, extending it to functions the earlier definition could not handle. Here the referent is a mathematical operation on functions, and it has nothing to do with a protein lodged in a cell membrane or a correcting term wired inside a feedback controller.[1]

Unit

One integral: a function paired with an interval or region, evaluating to a number when definite or a function when indefinite

Recovery means

Not applicable

Measured by

Computed as a limit of Riemann sums, by antidifferentiation through the fundamental theorem, or numerically by quadrature rules

An integral membrane protein is a protein permanently lodged within a cell's fatty membrane, held in place by stretches that cross the oily interior of the bilayer, so that it cannot be freed without breaking the membrane apart with detergents. Singer and Nicolson's 1972 fluid mosaic model, published in Science, set these proteins apart from peripheral membrane proteins, which merely rest on the surface and rinse off under mild conditions; the word integral here means built in and inseparable, an essential part of the structure. This referent is a physical molecule anchored in a living membrane, and the word carries a completely different meaning from the accumulating operation of calculus or the offset-correcting term of a controller.[2]

Unit

One protein species that spans or is embedded in the lipid bilayer, counted per membrane

Recovery means

Not applicable

Measured by

Identified by resistance to removal except with detergents, by hydropathy plots that flag membrane-crossing segments, and by structures solved with X-ray crystallography or cryo-electron microscopy

In a feedback controller, the integral term is the control action that accumulates the running error between where a system is and where it should be, and keeps pushing until that stored-up error is worked off. Minorsky's 1922 study of automatic ship steering and Åström and Hägglund's 2001 review The future of PID control describe how this term, the I in a PID controller, removes the steady offset that proportional action alone leaves behind, at the cost of a lag that must be tuned and guarded against wind-up. The referent is an engineered part of a control law, scaled by an integral gain, and although it is literally built from the calculus integral, it is a distinct working object, and it is unrelated to a protein fixed in a membrane.[3][4]

Unit

One integral term inside a control law, scaled by an integral gain or, equivalently, an integral time

Recovery means

Anti-windup limits the stored error when the actuator saturates, so the term does not overshoot as the system recovers

Measured by

Set by tuning the integral gain, often stated as a reset or integral time, using rules such as those of Ziegler and Nichols