AP Calculus AB / Mr. Hansen 5/17/2005

Name: ________________________

Polar Arc Length Worksheet

1.	Write the arc-length formula that was given in class for any 2-dimensional curve defined parametrically. You may need to consult your class notes.
2.	Recall from Precalculus that any polar curve can be converted into rectangular form by means of the equations x = r cos q, y = r sin q. Make a sketch to illustrate why these equations are true.
3.	Use the equations in #2 to develop an arc-length integral formula that works for polar curves, i.e., curves in which r is expressed as some function of q. Show your steps. (Hint: Treat q as playing the role of the parameter t, and treat x, y, and r as being functions of q, i.e., t.) The algebra, while straightforward, consumes a fair bit of space and may require a separate sheet if you write large. When you have finished, consult a textbook or a Web source (such as Will Felder’s BC Cram Sheet) to verify that you found the correct integral formula.
4.	Sketch each of the following polar curves. Then, use the result of #3 to find the arc length by the integration method, and use a secondary method to verify that your answer is correct or at least reasonable.
(a)	r = 2
	Sketch:
	Arc length by integration:
	Secondary method / reasonableness check:
(b)	r = 3 cos q
	Sketch:
	Arc length by integration:
	Secondary method / reasonableness check:
(c)	r = 1 + sin q
	Sketch:
	Arc length by integration:
	Secondary method / reasonableness check: