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Equation 10 · Part 8 · The Machine Before Electronics

fraction

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

What this part means

Divide the expression above the line by the one below it.

Its job in the formula

The expression above the fraction bar is divided by the complete expression below it. The denominator must not be zero.

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.

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

Open the illustrated fractions: division written vertically guide →

Sources cited in the article section

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