Integrating simple limits
Solve the integral
∫ − 1 1 n → ∞ lim ∣ ∣ ∣ n 1 − x 2 ∣ ∣ ∣ d x
The answer is 2.
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1 solution
lim n → ∞ ( x n + y n ) = 1 is one way to "square a circle". :)
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Haha, nice little "punn" (I guess you could call it that)
I remember reading up on that topic a while ago and how it was proven to be impossible just over a century ago.
Do you know how to "properly" integrate this function? Or is the integral just simply 1?
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Ooops.... make that lim n → ∞ ( x 2 n + y 2 n ) = 1 .
Anyway, for the integral I would let x = sin ( θ ) ⟹ d x = cos ( θ ) d θ . Then we would have
∫ n → ∞ lim cos ( 1 + n 2 ) ( θ ) d θ = ∫ cos ( θ ) d θ = sin ( θ ) = x
evaluated from − 1 to 1 , yielding the answer of 2 .
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@Brian Charlesworth – Oops, I made an error as well. I put 1 instead of x . 1 can't possibly be the integral of a function since it's a constant XD .
This is going on my list of "Dumbest things I've said" hanging on my bedroom wall just below "Does the Pythagorean Theorem work for all right triangles?
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@Trevor Arashiro – Haha. Well, 1 can be the solution of a definite integral. And as for the Pythagorean Theorem question, what you probably had in mind was whether or not the Theorem worked for spherical right triangles. The 2-D version is actually a special case of the 3-D version
cos ( R c ) = cos ( R a ) cos ( R b ) ,
where the 2-D version is obtained by letting R go to ∞ , (and messing around with some handy dandy Taylor series).
No question is "dumb" when looked at from the right angle. :) (And no, I can't resist a good pun once it presents itself.)
P.S.. In anticipation, yes, in the case of hyperbolic geometry we would have the corresponding formula cosh ( c ) = cosh ( a ) cosh ( b ) .
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@Brian Charlesworth – Yes, I was thinking of triangles that existed in hyperbolic geometry as well as in higher dimensions.
At the time, I had no idea that spherical geometry even existed.
......guess I wasn't that "well rounded" back then.... Now look what you started :P
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@Trevor Arashiro – Arghhh... Sorry, but yeah, there's something complementary about math and puns, and that's not hyperbole. O.k., I'll stop now before the punning reaches a critical point. :)
Here's a visual:
I the image doesn't work, here's the link .
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I made it.
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@Daniel Liu – WOW! How did you make this? Did u use geogebra? Because I've been looking for a simple way to do this as im not too well oriented with the more complex applications such as Adobe.
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@Trevor Arashiro – I took a bunch of screenshots of the graphs I made on desmos.com. Then, I searched up "how to create an animated GIF", and then clicked on the first link (imgflip.com I think). Then I uploaded all the images, set the speed to what I wanted it to be, and there you have it.
I tried to make it display by downloading and reuploading it on imgur, but even that doesn't work for some reason.
Ya forgot yer d x there ;p
∞ n = 1 assuming n = ∞ , 0
So we have a straight line with a single "hole" at (0,0)
This is like having a rectangle of height 1 and width of 2. So the area of this rectangle is 2.
The main reason why I posted this is because I'd love to see a solution that actually does this by proper integration because wolfram "couldn't interpret my input" because it's too complex.