Exam 1 Wed Feb 1 7:00pm in Pepper Canyon 109 (not 106!! different class there!)
Office hours: 2:45pm-4:15pm Next: Complex numbers (appendix G); complex exponentials (supplement, which is freely available online). We will not do arc length.
People were most confused last time by plotting curves in polar
coordinates. (1) it GOAL for today: Integration in the context of polar coordinates. Get much better at working with polar coordinates! |

Multiplying both sides of the equation by yields

But it isn't... if we remember the basic idea of calculus: subdivide and take a limit.

[[Draw a section of a curve for in some interval , and shade in the area of the arc.]]

We know how to compute the area of a sector, i.e., piece of a circle with angle . [[draw picture]]. This is the basic polar region. The area is

(fraction of the circle) (area of circle)

We now imitate what we did before with Riemann sums. We chop up, approximate, and take a limit. Break the interval of angles from to into subintervals. Choose in each interval. The area of each slice is approximately . Thus

Area of the shaded region

Taking the limit, we see that