Since the beginning of this series of infinitesimal
calculus we have gathered a lot of information on formulating the physical
world around us. Going through these will help you understand further.
So in previous post we have seen tangents and how to
find derivative of a function using some mathematical operations. Now we will
talk about ‘Problem of Areas’. What is area? It’s just a region in 2
dimensional space. We can find the area of square, rectangle, angle, etc. This is very
easy because we know their dimensions of length and breadth. What about circle
and ellipse? How do we know that area of circle is pi times the radius squared?
In the second part of calculus we will figure out how
to find area under a given curve. So to begin with let’s take an example of
curve y=x2. So we need to find the area under this curve. And to do
so we can use some figures whose area is known to us. In this case let’s say we
take 2 rectangles and draw them in the curve. As you can see lot of area is
left, hence draw 3 rectangles. Now we covered up a little more area, so if we
continue to add rectangles we will get closer and closer to actual area. In
order to find the accurate area we need to assume there are infinite amount of
such rectangles in between.
So there can be 2 cases, one where the rectangles are
taken under the curve and one taken above. But the actual area should lie
somewhere between these two values. This is famously known as the 'Sandwich Thermo'. Because the actual function is sandwiched between 2 functions.
This process of finding area under curve is called
‘Integration’. A fancy word for saying: find the sum of all the area units
under a curve. As the sum consisting infinite area units, it’s represented by a
stretched out letter 'S'.
But doing this graphically for all the functions is
nearly impossible so algebra comes to our rescue. We can use variables to
represent the functions. And here by, we introduce to the first fundamental
theorem of calculus. This theorem links two beautiful ideas in calculus,
integration and differentiation. In simple terms integration is a reverse
process of differentiation, and vice-versa!
Hence now you might be thinking what does all this has to
do with the actual physical world. And it turns out we can formulate the things
we observe and explain them using this mathematical tools as a language. These
mathematical concepts are a connection between us and the universe! So now it’s
up to us how better can we be at understanding the surrounding.
#B.R.A.I.N Foundation.
/Credits-- Piyush Suryawanshi6719
Stay tuned for more such mind baffling ideas...
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