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Homework Statement
Wikipedia tells me that I can obtain the surface area of a sphere by realizing that the volume of a sphere is equivalent to the infinite sum of the surface areas of hollow, nested spheres, sort of like little Russian dolls. That makes sense, and then differentiating both sides immediately yields the usual formula for the surface area of a sphere.
However when I attempt to use this approach for the lateral surface area of a right circular cone it produces an obviously wrong answer. I'm assuming we can similarly identify the volume of a cone as the infinite sum of the surface areas of hollow, nested cones that 'sit' inside the overall cone.
Why is this approach not fungible in this case?
Homework Equations
Case of the sphere:
V = 4/3 πr3 = ∫ SA dr where the integral is over 0 to r, and SA is the function of r that represents the surface area of a sphere of a particular radius,
and therefore
dV/dr = 4πr2 = SA(r)
The Attempt at a Solution
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Case of the right circular cone:
V = 1/3 πr2h= ∫ SA dr, holding h constant, where the integral is over the base of the cone from 0 to r, and SA is the function of r that represents the lateral surface area of a sphere of a particular radius,
and therefore
dV/dr = 2/3πrh = SA(r), where obviously it should be πrl, where l2 = (r2 + h2)