2008-01-15

Today's nonsense
- Explained to Ketteman, Bill and Spiller that the area of a circle can be found by treating it like a triangle with an area of (1/2)bh where we understand that b and h are equivalent to r and therefore we have r^2 and then we multiply it by 2 pi because that's the number of rotations, and therefore the area of a circle is pi * r^2
- Homework due tomorrow. And today.
- Tomorrow: pg 725 #38,40,42-52 setup and solve.

Arc length

How long is that curve? We cannot use the distance formula because it's not linear. The length of a curve can be figured out as a function. Recall, we're going to recall, that arclength for f(x) on [a,b] is ... imagine that the curve is an infinte number of little small line segments, so you're doing them and then you have an infinite summation which is an integration problem, so we found that S (for arc length) = integral from a to b of sqrt(1 + [f'(x)]^2) dx ... and that would get you the length of a curve.

S = integral from a to b of sqrt(1 + [f'(x)]^2) dx

Arc length of a parametric curve

For parametrics, S = integral from a to be of sqrt(1 + (dy/dx)^2) dx
= integral from a to b of sqrt(1 + ( (dy/dt) / (dx/dt) )^2 ) dx
= integral from a to b of sqrt( ( (dx/dt)^2 because that's what you need to make it that original '1' because dx/dt squared over dx/dt squared will give you the one + ( (dy/dt)^2 ) ) / (dx/dt)^2 ) dx
= integral from a to b of sqrt( ( (dx/dt)^2 + ( (dy/dt)^2 ) ) ) dt
Which looks suspiciously like the Pythagorean theorem.

So when we're trying to figure out the concavity of a segment, find the second derivative of the parametric equation, and then we plug in a value any value less than zero into the second derivative, and then another one from > 0 on the other side. So if it's - from -infinity to zero, then it's concave down on that interval, and if it's positive from zero to infinity then it's concave up on that interval. If there's multiple inflection points, you have to do this test mutiple times. The only difference here is on how to find the second derivative, otherwise it's exactly like last year.

Remember to watch out for domain restrictions, like if your x or y equation in parametrics has a natural logarithm or something (where the domain must be > 0)