009C Sample Final 1, Problem 9

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A curve is given in polar coordinates by

Find the length of the curve.

Foundations:  
1. The formula for the arc length of a polar curve with is
2. How would you integrate
You could use trig substitution and let
3. Recall that

Solution:

Step 1:  
First, we need to calculate .
Since
Using the formula in Foundations, we have
Step 2:  
Now, we proceed using trig substitution. Let Then,
So, the integral becomes
Step 3:  
Since we have
So, we have
Final Answer:  
  

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