I’ve been fascinated by the mental being of integral since lukio (high school in the US). It’s fascinating, for example, to see how it increases the dimension: From length we get into area, from area we get into volume.
Here I deduce a formula of circle’s area starting with length of a circle and the volume of a sphere starting from area of a circle by integrating.
The length of a circle is 2πr. By integrating we get into the next dimension: Area.
What is to be integrated is half of the length of the circle due to the way of determining the area. So we will be integrating πr.
We will be using the definite integral with –r as bottom limit and r as the upper limit, so that we get the area of whole circle.
Let’s take a look at the magic of integral:
The area of a circle is πr2 , what you can see as a result above.
We can use the integration formula of even function, because the curve behaves like even function’s curve.
Next to the volume of a sphere from area of a circle. We will be integrating ½πr2; as half of the area rolls over, we get a sphere, which is what we’re after.
The integration limits are now from 0 to 2r. Let’s see the magic:
As we remember, the volume of a sphere is 4/3 πr3. See the result above.
In this context it is fascinating to try to see the mental being of integral, how it increases the dimension. When you really understand the definition of integral you really can appreciate the mental being of integral.
I apologize that you are forced to read my terrible handwriting. 🙂
(Ooops! I should’t have used the r as variable and as the limits of the definitive integral, the variable could be r‘. Anyway, the result is the same.)