- You have an assignment due soon (well, today actually). Many of
you have figured out that I'm not a hard task-master on these
deadlines. Get me it by tomorrow and I'll be happy.
- Today there's a set of four (short) talks, by undergraduates (3:05, MEP461): Come if you can!
- We finish up Sect. 10.4: Polar coordinates (pdf, pdf summary)
In particular, we want to talk about area of polar curves, and turn polar equations into parametric equations in Cartesian coordinates.
Let's start with a circle of radius r, centered at the origin, and see how we might adapt the "familiar formulas" to more exotic shapes:
- Length: The arc length of a sector of the circle swept out
by angle $\theta$ is given by $r\theta$ (with the familiar
circumference formula being the case of sweeping out an angle
of $2{\pi}$.
So a small (infinitesimal) change $d\theta$ would sweep out a length of $rd{\theta}$.
- Area: The area of the sector is basically that of a wedge, which we can think of as a half rectangle. Let's compute its dimensions (Figure 3, p. 709).
- Length: The arc length of a sector of the circle swept out
by angle $\theta$ is given by $r\theta$ (with the familiar
circumference formula being the case of sweeping out an angle
of $2{\pi}$.
