We all know that the area of a quarter of a circle of radius 1 is
. However, I bet you didn’t know that apart from using geometric tools, there is another more ingenious and astounding way of calculating 
. A method so great that it combines integration, trigonometry and series expansion and the man responsible for it, Newton’s arch rival Leibniz. Let’s see how he did it.NOTE: I honestly did my best to minimize the amount of videos but it still came to three in the interest of maintaining the mathematical rigor in the derivation. Hope you watch all three.
LEIBNIZ'S QUEST FOR PI.3
May want to check out:
LEIBNIZ'S QUEST FOR PI.1
LEIBNIZ'S QUEST FOR PI.2
WALLIS'S PRODUCT
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VIDEO LECTURE part 3
MAIN CONCEPTS
Using techniques of calculus, trigonometry and series expansion,  Get the Printer Friendly VersionCOMMENTS
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LESSON
The area we are concern with is area A of the circular segment cut off by the chord OT as illustrated below. By finding this area, we simply add the area of the triangle which is
 to get the area of a quarter of a radius 1 circle.
While you may be busy trying to find how we will apply the Riemann sums method in finding the area, here is what Leibniz did. He considered the area of OPQ, the rather sliverlike elements whose base is the segment PQ of length ds. To get the area of a single element, we extend PQ to the area around O and then drop a perpendicular from PQ at R to meet O, thus OR represents the height of triangle OSP. A graceful play of geometry indeed.
(For a quick explanation, just shift the angle at P to the y-axis and you’ll see that this is the same angle at O of triangle ORS, bearing in mind that R is a right angle.) We now represent a small area of sliver element OPQ as dA giving us
Here, y denotes the length of the segment OS. The element of area dA sweeps across the circular segment as x increase from 0 to 1, what is initially wanted to find. So,
Anticipating that it will be easier to integrate a function of x w.r.t y, we use integration by parts to switch the variables.
where the limits of integrals are y = 0 and y = 1. Now it is here where we seem to be stuck. The idea is that we must find an equation of x in terms of y so that we can do the necessary integration. This poses a difficult problem as y is hard to define algebraically. Here comes Leibniz with his idea of using trigonometry. From the diagram, notice that
(If you need convincing of the angles We now employ some trigonometry identities and write,
which nicely gives us
our key in doing the integration. But wait, there’s still a little more work to be done. Recall a little formula that enables us to expand as an infinite geometric series.
which allows us to write
and substituting this into our calculation of the area A,
We add
which is a fine testament of a great mathematician linking different branches of mathematics together to find an irrational number. With this discovery, Leibniz undoubtedly won the battle over Newton in this area. All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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and thus 
and 
and 
, just observe that the angles at P and O are both right angles and so a straight line bisects that ‘kite’ shape.)
to this area to account for the quarter triangle of the circle. The result is equated to the known area of 
and so we have Leibniz’s formula
