What I want to talk about now is something related to geometry, trigonometry, and calculus, namely the area of a circle. Most people that have taken basic geometry in grade school can tell you the area of the circle is πr^2, but ask the average person why that is so and they wouldn't have a flippin' clue. I'm about to prove how exactly you solve for the area of a circle using analytical geometry and calculus.
In Figure 1, we see the geometry of the circle. The variable r is the radius of the circle, a is any arbitrary distance from the center of the circle, L is the length of the chord the distance a away from the center of the circle, and da represents a minute thickness of the segment dA with length L that contributes to the overall area of the circle. The summation of all of the dA’s (horizontal chords of minute thickness) is the exact area of the circle.The area of the segment dA is L da. We know by the Pythagorean theorem that the length of L is twice the length sqrt(r^2-a^2). Thus:
To find the summation the total area of the circle, we will sum up all of the dA’s from a=−r to a=r:
This being a complex integral, the use of trigonometric substitution is necessary for solving it. In figure 2, the triangle is shown in more detail. The value for a could be replaced by a trigonometric equality, a=r sin θ.The integral therefore becomes:
One of the essential trigonometric identities we come across is 1-(sin θ)^2=(cos θ)^2. Since there are mixed variables in the integral, we must somehow make a substitution. A correlation between da and dθ can be determined by the equation a=r sin θ.
When changing the variable, it is necessary to change the limits of the integral to correspond to the new variable. To find the value the limits change to, we solve for θ:
To complete the integration, we must recognize another trigonometric equality, namely
(cos θ)^2=(1 + cos 2θ)/2. We then derive:
(cos θ)^2=(1 + cos 2θ)/2. We then derive:
And there you have it, the area of a circle. You might have thought it would be a little more simple than that, but you were wrong.
So, I realize I just made myself look like a nerd, but I don't really care, because I just had a great time doing it. The end.
12 comments:
So, how do you calculate the volume of an egg?
Vater
This is how you would do it:
1. You first trace the egg onto grid paper. The entire length of the egg will be along some x axis, which bisects the egg into two parts, and one end of the egg touches the y axis.
2. Measure precisely multiple thicknesses of half of the egg along the x axis.
3. Determine a function f(x) for the curve that fits the thickness measurements calculated. There are programs that can do this for you, for instance, Logger Pro.
4. Sum up all the volume of vertical washers of minute thickness and radius equal to the curve function along the entire length of the egg. This requires an integral.
That's how you calculate the volume of an egg.
Or you could simply submerse it in water and record displacement. That method's too easy though.
Perhaps I should explain the procedure in better detail in a later post.
congratulations, Jonny. your blog put me to sleep. Honestly, math gives me a headache. I haven't touched it since college freshman year, with the exception of accounting, which wasn't too bad. Can you believe I actually wanted to be a math teacher at one point???
~SARAH~
Ok, so if I'm reading correctly, first we.... OWW... OOWWWW!!! ::hold head, while wincing::
jonny.. most people blog about normal stuff.. but you blog about something most of the world wont understand. maybe you should have a summer break! :D
Jonny... you make us wait 5 months for a new post, and this is what you give us? Circles? Wow.
...oh Sarah, you think that's sleepy, have you seen your husband's latest post? ;)
i always liked the method of cutting the circle up into little "triangles" with a bit of an arc on the base. then finding the area of the triangles and finding the limit as the number of triangles goes to infinity. a little easier to picture for me.
Good grief, Jon!!! It's about time you posted something about your incredibly interesting life!
Yeah, I'm going to have to apologize to my readers for making their day just a little more boring. I would gladly refund you for the time you could have been spent doing something more constructive (such as watching the latest episodes of “Will It Blend” on YouTube, reading about the newest breaking developments on the life of Paris Hilton at CNN.com, or perhaps making long, personal, viral MySpace surveys). I'll find some other forum to exploit my world of mathematical fantasies.
jonnny.. will it bend is so cool!! i could watch that all day.... or an i just further proving my pathetic life>?
There was a hillbilly who sent his son off to college and after the first simester the son came home and his dad asked him what he learned. Well Paw, I learned pie are squared. Why ya dummy his dad sparked off. Pie ain't squared, pie is round!
Simple Simon met a pi man
Going to the fair.
Said Simple Simon to the pi man
"You have unusual ware.
The (pi)s I've seen before were round
But gosh, you're (pi)s r^2."
Mathematical Nursery Rhyme No. 3
I am glad to see that you like math. Have you seen the circle area derivation based on taking the limit as n goes to infinity of the area of an n-sided polygon?
Dennis
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