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Happy Tuesday! Any questions about the tangents to polar curves assignment?

Topic of the day: Area of Polar Curves

Remember in calculating areas "under the curve" in AB, we were calculating the limit of an infinite sum of rectangles (because we were in a rectangular system!). You'll have to get rid of the up-down, right-left, outer-inner mindset of rectangular areas.

In polar, we're making slices of pi/pie (a much happier metaphor and appropriate to the holiday season). Instead of rectangles, we use the area of a sector: $LaTeX: A=\frac{1}{2}\cdot r^2\cdot\theta$

How do we add up an infinite number of pieces of pi? Integration, of course:

$LaTeX: A=\int\frac{1}{2}\cdot r^2\cdot d\theta$

The concept is straight-forward. What most people struggle with is the limits of integration (which are angles $LaTeX: \theta=a$ and $LaTeX: \theta=b$ .

One curve is usually simple enough, but if you are doing two curves, you'll need to find intersections (set the two curves equal and solve for $LaTeX: \theta$ ), and you need to have a good sense of where curves "start" and how they develop (thus my insistence on graphing). Always make use of symmetry when possible to simplify your calculations (even if you're using a GC).

Here's a great visual of why it isn't "outer minus inner":

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Homework:

10.7.1: p. 702 #1 - 15 odd. Keep drawing the graphs, especially doing area problems. It will help you understand the intersections better, the ways you can use symmetry to simplify the problem, and where to set your limits of integration.

Homework Solutions

Things you MUST KNOW:

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

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

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

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

Video Supports for Asynchronous Learning:

Patrick Math:

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OCT:

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Quotation Page

xkcd seasonals

Lin Christmas Lights