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Question
Find the area of the surface generated by revolving the given curve about the x-axis.
x=t
y=t^3 /3     for t=0 to t=3

Answer
This parametric eqn is the same as the eqn y = (x^3)/3. Since we are finding the surface of revolution, S, about the x axis, the function y = f(x) gives the radius of the circle around the x axis, at a particular x value. The revolution abround the x axis is thus a circle of circumference 2πy (y = (x^3)/3).

At each point x along the curve there corresponds an infinitesimal length, dl, centered at x, that gets revolved around the x axis, i.e, it gets multiplied by 2πy. It thus sweeps out a circular ribbon of width dl in a complete circle. We then need to integrate the slices along the x axis from 0 to 3, i.e., calculate ∫2πy・dl from 0 to 3

The length dl is given by the so-called arclength of the curve y = f(x) from x to x+dx. Just like a right triangle, this length is dl = sqrt( dx^2 + dy^2). This can also be written as

dl = sqrt(1+(dy/dx)^2)・dx.

Calculating df(x)/dx and plugging it, f(x) and dl into the integral ∫2πy・dl gives

S = ∫ (2π/3) (x^3) ((1+x^2)^1/2) dx integrated from 0 to 3.

This is the formula you need to calculate to solve for S given the problem description. Please let me know if you understand (and buy) the above derivation. If you need help evaluating the integral, let me know. Good luck.

Randy

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randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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