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

• charlie95
In summary, to find the volume of a football with a radius of 1 meter, first try to solve the problem using integrals, and then measure the volume or mass of the displaced water.
charlie95
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 )

American football or rest-of-the-world football?

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.

## 1. How do you calculate the volume of a football using integration?

To calculate the volume of a football using integration, you will need to use the formula V = ∫πr²dh, where V is the volume, π is pi (approximately 3.14), r is the radius of the football, and h is the height of the football. You will also need to use the equation for the surface area of a sphere, A = 4πr², to find the value for r. By integrating the surface area equation, you can find the volume of the football.

## 2. Why is integration used to calculate the volume of a football?

Integration is used to calculate the volume of a football because it allows us to find the volume of an irregularly shaped object, such as a football, by breaking it down into infinitesimal parts and summing them up. This is known as the "sum of slices" method, which is commonly used in calculus.

## 3. What materials are needed to calculate the volume of a football using integration?

In order to calculate the volume of a football using integration, you will need a football, a ruler or measuring tape to find the radius and height of the football, and a basic understanding of calculus and integration.

## 4. Can the volume of a football be calculated using other methods besides integration?

Yes, the volume of a football can also be calculated using other methods such as the "cavalieri's principle" or by using the formula for the volume of a prolate spheroid. However, integration is often the most accurate and efficient method for calculating the volume of an irregularly shaped object like a football.

## 5. Are there any real-life applications for calculating the volume of a football using integration?

Calculating the volume of a football using integration has real-life applications in fields such as sports science and engineering. Knowing the volume of a football can help with designing more efficient and aerodynamic footballs, as well as understanding the physics behind how the ball moves and interacts with players during a game.

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