Using Pappus' Theorem to Find Volumes of Solids of Revolution

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Homework Help Overview

The discussion revolves around applying Pappus' Theorem to find the volumes of solids of revolution, specifically focusing on the region under the graph of y = x^3 from x=0 to x=2 when rotated about the y-axis. Participants are addressing various aspects of the problem, including area, volume, and centroid calculations.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the calculation of the area of region R and the volume of the solid using different methods, including vertical slices and the method of shells. Questions arise regarding the correctness of the volume calculation and the steps taken to arrive at those results.

Discussion Status

There is an ongoing exchange where one participant expresses uncertainty about the volume calculation, prompting requests for clarification and a more detailed explanation of the methods used. The discussion is active, with participants seeking to understand each other's reasoning and calculations.

Contextual Notes

Participants are working within the constraints of the problem as posed, which includes specific tasks related to the application of Pappus' Theorem and the calculations of area, volume, and centroid. There is an emphasis on ensuring clarity in the methods used to solve each part of the problem.

Pirata
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The Region "R" under the graph of y = x^3 from x=0 to x=2 is rotated about the y-axis to form a solid.

a. Find the area of R.
b. Find the volume of the solid using vertical slices.
c. Find the first moment of area of R with respect to the y-axis. What do you notice about the integral?
d. Find the x coordinate of the centroid of R.
e. A theorem of Pappus states that the volume of a solid of revolution equals the area of the region being rotated times the distance the centroid of the region travels. Show that this problem confirms this theorem.

The Attempt at a Solution


I was able to do part "a" as the integral from 0 to 2 of x^3 dx. Also I believe part "b" is pi*[3y^(5/3)/5] evaluated from 0 to 2
 
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I don't think "b" is right. How did you get that?
 
Well I posted what I got. Was hoping you would show me where I went wrong...
 
It's best if you show how you got what you got. Otherwise we just have to guess how you got what you got. That's not fun. Use the method of shells to find the volume. V=integral(2*pi*x*height*dx).
 

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