Tasks for today:

Good luck on your other finals and projects
Rest, relax, and review if you think it's necessary (I don't think it is -- I would rather you got a good night's sleep!)
Finish any college apps due December 1st!

Review information:

Arc Length of Polar Curves:

Connecting the dots with arc length (pun intended).

Using the distance formula, a length is

$LaTeX: L=\sqrt{\left(\Delta x\right)^2+\left(\Delta y\right)^2}$

If you want the length of a curve, you add up a bunch of tiny little lengths:

$LaTeX: L=\int\sqrt{\left(dx\right)^2+\left(dy\right)^2}$

But we used a little algebra (factoring out the LaTeX: dx to simplify the formula):

$LaTeX: L=\int\sqrt{1+\left(dy\right)^2}\cdot dx$

Then we did parametric curves, and the arc length formula looked more like the distance formula again because our variable of integration was t :

$LaTeX: L=\int\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\cdot dt$

So now that we're thinking about our independent variable as $LaTeX: \theta$ , we might expect the arc length formula to look like:

$LaTeX: L=\int\sqrt{\left(\frac{dx}{d\theta}\right)^2+\left(\frac{dy}{d\theta}\right)^2}\cdot d\theta$ AND IT IS! . . . but it's simplifies to:

$LaTeX: L=\int\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\cdot d\theta$

Homework:

Things you MUST KNOW (short version):

$LaTeX: y=r\cdot\sin\left(\theta\right)$

$LaTeX: x=r\cdot\cos\left(\theta\right)$

LaTeX: x^2+y^2=r^2

$LaTeX: \tan\left(\theta\right)=\frac{y}{x}$

Video Resource:

From OCT