Hello, guys! It’s me! Warotan!

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This time, let’s calculate the volume of a 3 dimensional ball toward that of an \(n\) dimensional ball.

Also, my misunderstanding about study in high school is given.

I drunk too much today, so there might be some mistakes.

The year-end party was so much fun!!

I would like to tell you the construction of this page and please select where you start to read to your interests.

## The volume of a 3 dimensional ball

We know that the volume of a 3 dimensional ball is written as

The suffix 3 shows that it is a THREE dimensional one.

How can we derive this formula?

Let us first consider a semicircle as a preparation for the 3 dimensional volume.

If we rotate this around the \(x\) axis, we can make a three dimensional ball.

The area of a circle created by turning the blue line at \(X\) around the \(x\) axis is given by \(\pi \xi^2\), \(\xi \equiv \sqrt{r^2-X^2}\).

If we consider a disk of thickness \({\rm d}X\), the volume is \(\pi \xi^2{\rm d}X\).

Summing up this volume in \(X\in\left[-r,r\right]\) gives us the desired volume;

It is a kind of shell integration.

## Story of my school life~Integration is a method of calculating the area of a graph!?~

In addition to the main story, I tell you my mistake about study.

When I was a high school student, I was not (and am not so) good at studying.

One of my big misinterpretation was about integration.

In those days, I thought that integration is a method of calculating an area of a graph.

Exercises about integration given by the school was all to calculate an area such as

“calculate the area between \(y=-x^2+3\) and the \(x\) axis” or

“calculate the area between \(y=\sin\left(x\right)\,\left(0\leq x\leq \pi\right)\) and the \(x\) axis”

and so on…

I came to think that

Ah… ok. Integration gives us an area!

BUT!!! That was a mistake.

After I passed the entrance exam of a university and, in the lecture,

a professor wrote some equations like

Here \(M\) is mass and \(\rho\) is density.

It confused me and I got into a panic.

“What does it mean??? Integration gives me an area but he says he calculates the mass of something???”

Of course it’s impossible for one who thinks integration is a method of calculating an area to understand the equation.

Now that makes me laugh but I was serious and asked him what the equation means.

He was surprised but taught me so kindly.

## Summary

Now, that’s all what I wanted to write.

This time, we did a warming up for the next step, volume of an \(n\) dimensional ball and I gave you a story of my school study.

Dividing a large problem into small ones is a usual practice (integration).

Thank you for reading and please spread this blog if you like.

If you have any comment, please let me know from the e-mail address below or the CONTACT on the menu bar.

tsunetthi(at)gmail.com

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