Calculating Volume of a Football Using Integration | Step-by-Step Guide

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Discussion Overview

The discussion revolves around calculating the volume of a football using integration techniques. Participants explore the mathematical setup required for this calculation, particularly focusing on the use of integrals and the challenges posed by the shape of a football, which is not a perfect sphere.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant asks how to calculate the volume of a football with a specified radius using integration.
  • Another participant emphasizes the need to set up the integral correctly before seeking help, suggesting a collaborative approach to problem-solving.
  • A participant notes that real footballs are roughly ellipsoidal, which complicates the mathematical modeling and suggests a practical method of measuring volume through water displacement.
  • One participant expresses frustration over the focus on the type of football and reiterates their interest in the mathematical approach, regardless of the specific object.
  • A later reply provides a formal expression for the volume integral but suggests switching to polar coordinates for convenience.
  • Another participant encourages using online resources to find existing explanations and suggests sharing specific difficulties for targeted assistance.
  • One participant claims to have solved the problem more easily by switching to polar coordinates, indicating a personal resolution to the inquiry.

Areas of Agreement / Disagreement

Participants generally agree on the need for a mathematical approach to the problem, but there are differing opinions on the best method to set up the integral and the relevance of the football's shape. The discussion remains unresolved regarding the most effective way to calculate the volume.

Contextual Notes

There are limitations in the discussion regarding the assumptions about the shape of the football and the mathematical expressions used. The complexity of integrating over the boundaries of a non-spherical object is acknowledged but not fully resolved.

charlie95
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How to find the volume?

If we have a footbal, let us say that the radius is 1meter, how do we calculate the volume ??
And show it with Integral! (V= ∫∫∫dxdydz )
 
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American football or rest-of-the-world football? :wink:

First, you try to set up the integral, and then we can help you improve it if it's wrong. We don't give direct answers to things like this here, but we'll try to steer you in the right direction towards the answer.
 


haha... funny guy :D rest-of the-world football!
If we have a huge footbal with radius of 1 meter. How could I find the volume of that ball without using a formulae book...
 


As jtbell said, we require that you first make an effort to solve it yourself and then we'll help you if you've done it wrong.
 


For any real football anything you can do mathematically will be at best a rough approximation. Unless you can come up with a precise mathematical expression. Footballs are roughly ellipsoid, however getting an exact expression may not be possible.

The best way to find the volume of a real object is not mathematically, but just dunk it in a container of water and measure the volume or mass of the displaced water.
 


just forget it. I was just interested to know how we can do this mathematically with integral. I don't care if it is a football/baskeball/tennsball or etc.. Just that it is round(sphere) and has a radius of 1 meter.. the radius is not that important either, it can be 1000000000000 meters... I am just interested to find out how we can calculate it mathematically... And this is not a task that I have been given...

I understand that russ watters. But i am not sure where to begin..
V=∫∫∫r dxdydz...
.
 


Well, formally, you have ##\displaystyle \iiint_{x^2+y^2+z^2<R^2}dx dy dz##. But that's not terribly helpful, because the integral boundary is inconvenient. Consider switching to polar coordinates. What are the dx, dy, and dz equal to in terms of dr, dθ, and dφ?
 


charlie95 said:
i am not sure where to begin...

Google might be of some help. Seriously! In the eight hours since your first post in this thread, you probably could have found many web pages that discuss finding the volume of a sphere via integration. (Yes, I've looked, myself, to make sure of this.)

If you have trouble understanding them, choose one, give us a link, and tell us what you don't understand about it. Then we'll have something specific to help you with.
 


I have a life jtbell...thanks for nothing...
I found many web pages, but many of them do it differently.thanks k^2... I solved the problem... much easier swithing over to polar coordinates.
 

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