Wright Math

Section 11.2 [12.2] really didn't involve calculus, so in this section we're getting back on the calculus train by learning about... 

• the slope of tangent lines for polar curves
• the area of a region bounded by polar curves

Tangent lines, Slopes, and Polar Curves

A really natural thing to think, when you have a polar function, is that the derivatiave of the function should give you the slope of the tangent line (isn't that what derivatives are, after all?).

That thinking doesn't work in this context, and here's a brief explanation as to why.

Slope is measured by the change in y with respect to the change in x.

Polar functions describe a relationship between r and theta, NOT x and y, so the derivative of a polar function tells us how r changes as theta changes, NOT how y changes as x changes!

This is important!  In Math 283, you'll be looking at points in a new way, using what's called a "Position Vector".  This will include analyzing  changes in the r-value as theta "sweeps" through its rotation.

Here's a video to help you visualize this "sweep".   As you watch, ask yourself how the r value is changing as the angle changes.  This rate of change is what dr/dtheta would measure.

Video:   "Sweeping" Polar Graphs

Okay, so we know what WON'T give us the slope of the tangent, so what WILL give us this slope?  It's still dy/dx, just as it always has been, and these next videos will show you how to find dy/dx for a polar function.

Video:     Slope of Tangent Lines

Vertical and Horizontal Tangents - Note: In this video, Krista shows a way of finding dy/dx without just plugging into the formula that Sal shows in the prior video.  Our book does it Sal's way.  Either method works!

Area of Regions

Now we're going back to the very beginning of the course, when we were determining area under and between curves.  In that situation, since we were using the Rectangular Coordinate system, our Riemann Sums consisted of adding up areas of . . . rectangles!  No surprise there.

The rectangles were formed by a little change in x (delta x) and the height given by the function.

In Polar Coordinates, we'll be looking instead at a little change in the angle, delta theta, and the "sector" that creates (a sector is just a slice of pie).  We'll sum up the area of all of those sectors that are formed as theta sweeps through a range of values and voila!  We'll have a sum that in the limit (as delta theta goes to 0) turns into an integral that gives the area of the region!

Again, for visualization purposes, to see the sectors that are created as you sweep through various ranges of angles, check out this Desmos tool.  The "a" and "b" are the limits of integration, and as you change the limits by moving the slider, you can see what region is sharded by all the little sectors.

Desmos Polar Shader

Now that you have the picture of what you'll be doing in mind, watch these videos to see what the area integral looks like and then how to apply it.

Video:  Introduction

Worked Example

Practice Setting Up Integrals - These problems will give you practice with identifying the limits of integration based on the curve and what region is shaded.

Assigned Problems:  You should include a graph with each of these problems, illustrating the curve and either the tangent or the area, as indicated.