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All of those (except the circle) have the 1/2 because they're integrals of something linear. While it's true that area and integral are closely related (the la
by amalcon 17y ago
All of those (except the circle) have the 1/2 because they're integrals of something linear. While it's true that area and integral are closely related (the latter being a special case of the former), a circle is clearly not linear.
- johnaspden 17y agoAs a circle expands, its area grows proportional to its circumference. The circumference is proportional to the radius. So you're getting the area by integrating a linear thing.
- deleted 17y ago[deleted]
- mhartl 17y agoAs a circle expands, its area grows proportional to its circumference. The circumference is proportional to the radius. Exactly. In symbols, this reads dA/dr = C ∝ r. Setting τ = C/r = 6.2831853…, we have dA = C dr = τ r dr => A = ½ τ r². Q.E.D.
- mhartl 17y agojohnaspden has it right. To put it more explicitly, we can calculate the area of a circle by integrating the differential element of area dA for an infinitesimal annulus from 0 to r. Now, dA is simply the arclength (circumference C) times the thickness dr; since the circumference scales linearly with radius, this leads to the integral of a linear function as follows: dA = C dr = τ r dr => A = ½ τ r².