Infinite Powers by Steven Strogatz

Author:Steven Strogatz
Language: eng
Format: epub
Publisher: Atlantic Books

The dotted line at x marks the current position of this imaginary roller as it rolls to the right. Meanwhile, to ensure that the region is painted neatly, the roller instantly and magically shrinks or stretches in the vertical direction, exactly as needed to reach the curve on top and the x-axis on the bottom without ever crossing those boundaries. The magical aspect is that it always adjusts its length to y(x) as it rolls, so it paints the area immaculately.

Having set up this far-fetched scenario, we ask: At what rate does the gray area expand as x moves to the right? Or, equivalently, what’s the rate at which paint is being laid down when the roller is at x? To answer that, think about what happens in the next infinitesimal interval of time. The roller rolls to the right through some infinitesimal distance dx. Meanwhile, as it traverses that tiny distance, it keeps its length y in the vertical direction almost perfectly constant, since there’s almost no time for it to change its length during the infinitesimally brief roll (a fine point that we’ll discuss in the next chapter). During that brief interval, it paints what is essentially a tall, thin rectangle of height y, infinitesimal width dx, and infinitesimal area dA = y dx. Dividing this equation by dx then reveals the rate at which area accumulates. It is given by

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