Area & Arc Length of $r=1+6\sin(\theta)+\cos(6\theta)$

Question

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.

Answer 1

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 $1/2\int r^2 \, d\theta$.

Answer 2

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