Volume of a cylinder, horizontally

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    Cylinder Volume
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Discussion Overview

The discussion revolves around finding the volume of a horizontally oriented cylinder using integration techniques, specifically when dividing the cylinder into rectangles rather than disks. Participants explore the challenges encountered in the integration process and the mathematical reasoning behind it.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant seeks to determine the volume of a cylinder of radius r and height h by integrating over rectangles, expressing difficulty in obtaining the correct integral results.
  • The participant notes that attempts have led to either the square root of a negative number or an integral that equals zero, indicating confusion in the integration process.
  • There is a mention of needing to incorporate π into the equation, with the participant considering the use of the Pythagorean theorem to define the width of the rectangles.
  • Another participant suggests starting with a simpler problem, such as finding the area of a circle using rectangular coordinates, to gain insight into the integration process.
  • A later reply reveals that a participant identified a missing trigonometric substitution that had been overlooked, which was crucial for solving the problem.
  • The participant expresses relief and gratitude for the advice, indicating that this realization has put them on the right track.

Areas of Agreement / Disagreement

The discussion reflects a lack of consensus on the integration approach for the volume of the cylinder, with participants sharing different strategies and insights without reaching a definitive solution.

Contextual Notes

Participants have not fully resolved the mathematical steps involved in the integration, and there are indications of missing assumptions regarding the setup of the problem.

trancefishy
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i posted a question earlier, but, the heart of the matter has been narrowed down. this question is much more straightforward than the details in the other one. what i want to know is how to obtain the volume of a cylinder of radius r and height h by integration.
EASY. if you divide it into disks and add them up. no problem. i need the volume when it is divided into rectangles. as in, the cylinder is lying on it's side.

try and try i might, i always end up with either the square root of a negative number, or the entire integral turns out to equal zero.

also, even though i know the integral should equal pi r^2 h, i don't see how to get pi into the equation. the only way i can see to define the width of the rectangles is by using the pythagorean thereom on a general triangle on the sides of the cylinder. any insight into this would be greatly appreciated
 
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Why don't you try something simpler first to get a handle on it? Try using rectangular coordinates to find the area of a circle.
 
hm, good thinking. i think I've done that before, but i will do it again. perhaps that will shed some light on this. this problem is just getting narrowed down thinner and thinner.
 
Last edited:
trancefishy said:
i need the volume when it is divided into rectangles. as in, the cylinder is lying on it's side.

try and try i might, i always end up with either the square root of a negative number, or the entire integral turns out to equal zero.

also, even though i know the integral should equal pi r^2 h, i don't see how to get pi into the equation. the only way i can see to define the width of the rectangles is by using the pythagorean thereom on a general triangle on the sides of the cylinder. any insight into this would be greatly appreciated

Show what you have done.

ehild
 
it took me a couple minutes to realize, this was the problem. i looked, and saw it was a simple trig substitution. i kid you not, i have spent, in the past 4 days, over 8 hours, solid, on and off, sometimes 2 hours at a time, wrestling with this problem. and it was a trig substitution the entire time that i was missing. that is why i couldn't integrate the cursed square root.

i feel like, overwhelmingly, stupid.

thank you very much for that advice, i am now on the right track. i wish i would have known this much much sooner.

thanks again
 

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