Math Is Fun Forum

Maybe the solution lies in the infinite pi-notation format of the reimann zeta function
After all, problems are a lot cooler to solve by hand than to approximate with computer

I believe the universe is a 256-dimensional space/time hyper-space that vaguely resembles the structure of a 256-dimensional hyperbola rotated around a 256-dimensional hypersphere perpendicular to the axis of the hyperbola. A warping of the space time continuum actually just shifts the foci in the 32nd, 64th, 96th, 128th, 160th, and 192nd dimensions.

Do I win?

(Where's my name in scientific American?)

0.03 seconds using the mathematica online integrator.

Isn't one of the bikini calculus girls an MIT graduate student or something?

But Geez... I downloaded one just to see if it's any good... and it doesnt teach jack

This works just like calculus, except you don't actually use an integral.

The strategy is to find the ratio of the volumes of a cone and it's respective cylinder, since once you find that, that remains constant no matter how you scale them.

Let the radius of the base = r, and without loss of generality (you'll see why), assume that r is an integer. Let the height = r also (to spare us of complexity)
We can approximate the volume of the cone by adding the volumes of r discs, where each disk has an integral radius from 1 to r.

(Ex, the first disk has radius 1, the next has 2, the next has 3 etc... the last has radius r)
Let the height of each disk be 1.
The volume is therefore:
summation of k as k ranges from 1 to r of

times the sum of the first r perfect squares

, by the formula for the sum of the first r perfect squares.

The volume of a cylinder with the same base and height is

The ratio of the area of the cone to the cylinder is therefore

Note that if we had used a height other than r, the height would've canceled out anyway once we divided by the volume of the cylinder.

Obviously, the more disks we have (the bigger r is), the more accurate the ratio will be.
As r gets very very big,

get's very very small, and

get's very very small.

At this point, note that since r is getting very very big, the assumption that it is an integer get's less and less important, because we are taking an infinte number of "samples".

So effectively, the ratio is

, and the volume of the cone is

the volume of the cylinder.

Volume of a cylinder is Area of base x Height, so the volume of a cone is 1/3 x Area of Base x Height

Again, these kinds of approximations work just like calculus and are the foundation of calculus, but you should be able to understand it without calculus.

PS - what is the latex tag on this forum? I'm sick of writing without latex...
PPS - Thanks John

Wikipedia article: Inflection point

I'm not really getting a square... at least not with the explicit equations.

The parametric square = awesome

The first derivative doesn't have to be 0 to have a PoI.

For example, the function f(x) = x^3 - 3x has inflection point when x = 0 because the following two conditions are met when x = 0:

f''(x) = 6x, which is 0 when x = 0
f'''(x) = 6, is not equal to 0 when x = 0

Note that although f'(x) = 3x^2 - 3 and f'(0) = -3, the function still changes concavity.

I guess you might as well just use Newton's recursion for these things...

Derivative =0

x^2 = 0, x = 0

or

2xlnxe - lnx = 0
lnx(2xe-1) = 0

lnx = 0 or 2ex - 1 - 0
x = 1 or x = 1/(2e)

So, horizontal tangents occur at
x = 1
x = 1/(2e)
x = 0

Double check that the second derivative is never equal to 0 for each of those points... when it does, it isn't an extrema

is log natural log or base 10?