← All parts of this equation

Equation 10 · Part 12 · The Machine Before Electronics

Ending index or upper bound: t

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

What this part means

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

Its job in the formula

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

The passage around this formula

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.

Read this part in the article →

Learn the underlying idea

Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.

Open the illustrated sums and products: repeat an operation over an index guide →

See this notation across published equations →

Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.