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How do I approach this problem?

Graph the following polar curve and show that it cuts out 5 different regions. Find the area of each of those regions.

$$r=1+6\sin(\theta)+\cos(6\theta)$$

Also, find the arc length (numeric value) for each of the loops of this (the unfolded) graph.

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What have you attempted? –  ncmathsadist Nov 22 '12 at 1:18
    
@ncmathsadist i figured out points where $r=0$ but how do i find the area of 5 different regions? –  jeffq Nov 22 '12 at 1:23
    
have you plotted a graph? –  ncmathsadist Nov 22 '12 at 1:26
    
@ncmathsadist yes i have –  jeffq Nov 22 '12 at 1:26
    
@jeffq: Have you got the answer yet? –  B. S. Nov 24 '12 at 9:11
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2 Answers 2

You need to look at which curve is outside and which is inside in each sector. Remember that area in polar coordinates is given by $1/2\int r^2 \, d\theta$.

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Here is the graph for theta in $[0,2\pi].$

enter image description here

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