You are here:

Advanced Math/Parametric Equations


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

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.


Advanced Math

All Answers

Answers by Expert:

Ask Experts


randy patton


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


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

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

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

Past/Present Clients
Also an Expert in Oceanography

©2017 All rights reserved.