Given information:

  r=1+2sinθ

Formula Used:

The formula for the area of the polar region:

  A=∫ab12 r2dθ

Calculation:

The equation of the region is

  r=1+2sin6θ

So, the curve region of r=1+2sin6θ is plotted as below:

Now, need to find the interval in which the inner loop exists:

  r=01+2sinθ=0sinθ=−12θ=7π6,11π6

That means the area lies between the interval 7π6 and 11π6 .

Since, the curve of the equation is symmetric about the polar axis. Os the area of the loop is gets double between θ=7π6,3π2

The area of given equation of the polar region is:

  A=∫ab12 r2dθA=∫ab12 (1+2sinθ)2dθA=12∫7π63π2 (1+4sinθ+4sin2θ)dθA=12∫7π63π2 (1+4sinθ+4×1−cos2θ2)dθ                            [∵sin2x=1−cos2x2]

  A=12∫7π63π2 (3+4sinθ−2cos2θ)dθA=12(3+4sinθ−2cos2θ)7π63π2                              [∵∫abcos2x dx=sin2x2∫ab sindx=−cosx∫ab1 dx=x]A=12(3(3π2)−0−0−3(7π2)+32−432)A=π−(332)