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Published equation contexts

z(t)=∫0ty(τ) dxdτ dτz(t) = \int_{0}^{t} y(\tau) \, \frac{dx}{d\tau} \, d\tau

Why this formula appears here

Summation is the easy operation to build. Integration is the one that opens up a whole class of problems, and the mechanism that made it practical was the wheel-and-disc integrator: a wheel riding on a rotating disc at a radius that can be varied, so that the wheel’s rotation accumulates the product of the disc’s rotation and the instantaneous radius. If the disc rotation represents the independent variable and the radial position represents the dependent one, the wheel angle represents z(t)=∫0ty(τ) dxdτ dτz(t) = \int_{0}^{t} y(\tau) \, \frac{dx}{d\tau} \, d\tau. which is to say, the integral of y with respect to x , produced continuously and without arithmetic.

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tt

Symbol t

t occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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τ\tau

Symbol τ

τ occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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00

Starting index or lower bound: 0

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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tt

Ending index or upper bound: t

This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

z(t)=∫0ty(τ) dxdτ dτz(t) = \int_{0}^{t} y(\tau) \, \frac{dx}{d\tau} \, d\tau

Equation 10 · History of Computing

The Machine Before Electronics

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Summation is the easy operation to build. Integration is the one that opens up a whole class of problems, and the mechanism that made it practical was the wheel-and-disc integrator: a wheel riding on a rotating disc at a radius that can be varied, so that the wheel’s rotation accumulates the product of the disc’s rotation and the instantaneous radius. If the disc rotation represents the independent variable and the radial position represents the dependent one, the wheel angle represents z(t)=∫0ty(τ) dxdτ dτz(t) = \int_{0}^{t} y(\tau) \, \frac{dx}{d\tau} \, d\tau. which is to say, the integral of y with respect to x , produced continuously and without arithmetic.

Meanings in this article

  • yy: the integral of.
  • xx: the integral of y with respect to.
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