#5 | Integral values can be calculated using only elementary school addition & multiplication! ∫f(t)dt in Excel cells

This article is part of the series here👇.

The goal of this series is to use
high school level math and physics to
theoretically animate the unnatural movement of a
10-yen coin in a Kokkuri-san session.

Last time toward this goal,
in order to express the movement of the 10-yen coin with formulas,
we created the framework of the equations of motion
to find acceleration in words.

Next, how do we derive
displacement (position) from that acceleration?

If you multiply acceleration by time,
you can understand the transition of velocity.

If you multiply velocity by time,
you can derive the transition of displacement.

I touched on this
last time as well.

To calculate these by hand,
you need to solve integral equations like the following.

$$
\footnotesize
\begin{align*}
（速度）=&\int（加速度）dt \\
（変位）=&\int（速度）dt
\end{align*}
$$

This acceleration
changes moment by moment.

Solving it is very tedious, isn't it?

So this time,
without doing difficult formula expansions,
we will think about how to find this integral value
using Excel cells as a calculator.

To you who thought,
I'll never use integrals in my life!🫵

If you read this article to the end,
you will grasp the basic concept of integration,
and you will be able to perform integral calculations
using only Excel cells!

Now, let's get started...

The integration learned in high school math
was a method to find values neatly using formulas.

However, computers
do not do that
when calculating integral values.

Its specialty
is being able to calculate at super speed.

Taking advantage of this specialty,
it calculates integral values as follows.

First,
the function to be integrated is
divided into very small intervals.

Here,
let the function to be integrated be $${f(t)}$$, and
assume it is divided at constant intervals of $${\Delta t}$$.

Then,
within this $${\Delta t}$$,
the Y-axis value (velocity or acceleration) is assumed to be constant.

By doing this, the integral value (area) of the curve
becomes approximately equal to the sum of the areas of thin rectangles.

In other words,
the integral formula learned in school, $${Y=\int f(t)\ dt}$$, is

$$
Vertical length f(t) × Horizontal infinitesimal width \Delta t
$$

and so,
the area of the thin rectangle is calculated by multiplication, and
as shown in the following formula, it is (approximately) replaced by the task of simply adding them all up.

$$
\Sigma \{Vertical length f(t) × Horizontal infinitesimal width \Delta t \}
$$

Therefore,
the integral formula can be solved by
repeating addition and multiplication a vast number of times.

By the way,
if you make this horizontal width $${\Delta t}$$ finer and finer,
and replace it with $${dt}$$ which is infinitely close to zero,
the area of the thin rectangle becomes $${f(t)\ dt}$$.

And,
the symbol $${\Sigma}$$ (sigma) for adding up separate rectangles changes its form into the symbol $${\int}$$ (integral) for adding them up smoothly.

With this,
the formula $${\int f(t)\ dt}$$ learned in high school mathematics is complete.

This time,
we considered the
integration method for finding displacement from acceleration.

From next time,
we plan to finally
calculate the displacement of the 10-yen coin.

Please look forward to it!

🔗 Continues to the next article

If you found this helpful or
are interested in the next part,
please give it a like or follow,
as it encourages me to create the next article!

• Subject: #Kokkuri-san

• Theme: #NumericalIntegration #EulerMethod

• Use Case: #IndependentResearch #Relearning #Relearning #ClassMaterial #Physics

• Topic: #ILovePhysics #IMadeThis