Surface of revolution - y = (1/3)x^3

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Discussion Overview

The discussion revolves around finding the area of the surface of revolution for the curve defined by \(y=\frac{1}{3}x^3\) from \(x=0\) to \(3\) about the y-axis. Participants explore various methods of integration and substitutions to solve the integral involved in calculating the surface area.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Exploratory
  • Debate/contested

Main Points Raised

  • One participant proposes an integral setup using \(y\) as the variable of integration, leading to a complex expression that involves multiple substitutions.
  • Another participant suggests integrating with respect to \(x\) instead, presenting several forms of the integral that could potentially simplify the calculation.
  • There is a question about the equivalence of different integral forms presented by participants, prompting clarification on their derivation.
  • Some participants express uncertainty about the anti-derivative of certain expressions, particularly relating to hyperbolic functions.
  • Discussions arise around the use of product rule and integration by parts, with participants attempting to relate their integrals to known derivatives.
  • One participant reflects on the difficulty of recognizing the product rule in the context of their integration approach.
  • Another participant shares their method of working through the integral, indicating a preference for a specific substitution technique.
  • There is a mention of the challenges faced when differentiating and integrating, with participants sharing their thought processes and frustrations.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method for solving the integral, with multiple competing views and approaches presented throughout the discussion. Some participants agree on certain steps but diverge on the overall strategy and techniques used.

Contextual Notes

Participants express uncertainty about the effectiveness of their chosen methods and the complexity of the integrals involved. There are unresolved questions regarding the equivalence of different integral forms and the appropriateness of various substitutions.

Who May Find This Useful

This discussion may be useful for students and practitioners interested in advanced calculus, particularly those exploring techniques for solving integrals related to surface areas of revolution.

  • #31
Rido12 said:
...
At the end of your solving, did you end up plugging $\tan^{-1}(9)$ in? Or did you convert the bounds to $z$? I've always avoided changing bounds to that of radians because of situations like this.

Yes, I used:

$$\theta=\tan^{-1}(9)\implies\tan(\theta)=9,\,\sec(\theta)=\sqrt{9^2+1}=\sqrt{82}$$

$$\theta=0\implies\tan(\theta)=0,\,\sec(\theta)=\sqrt{0^2+1}=1$$
 

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