As a word nerd, it was thrilling to learn that the word *parable* is related (metaphorically and etymologically) to the Latin word *parabola* + the Greek *parabole.* Online Etymology Dictionary explains that the sense of “juxtaposition” owes to the fact that the curve is “produced by ‘application’ of a given area to a given straight line.” So I’m digging around to learn more about the so-called *application of areas* and like, there’s a reason geometry was one of my least favorite subjects to study. It ought not be!
So, tell me — what is the application of areas? In what sense can an area be “applied”? To what? How the hell is this related to cones (and, as I’m seeing, squares/rectangles)?
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This is a question about calculus, and I’m really struggling to explain it in text. Integrals, and mathematical geometry in general, really benefit from visual aids.
The general example is the relationship between acceleration, velocity, distance, and time.
A car with constant acceleration will have a flat line for the acceleration graph.
To calculate the velocity graph, you measure the area beneath that flat line (acceleration x time) This produces a not-flat but straight line that grows indefinitely since you’re getting faster.
To calculate the displacement, you measure the area beneath that velocity line (velocity x time). This produces a curved exponential graph – you’re getting faster the further you go, so the chart experiences exponential growth.
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