Determine the area common to r= 3+3sintheta and r=3

Question

Determine the area common to r= 3+3sintheta and r=3

Answer 1

You should first graph these 2 functions...I suggest DESMOS for this.

The first equation graphs as a cardioid and you should look at the interval from 3π/2 to 2π.

You need to compute the area between the cardioid and the x-axis using an integral in polar co-ordinates.

If you don't know how to do that, you should look it up in your calculus textbook.

Then use the symmetry of the graph to give your answer.

Answer 2

If you polar plot the two curves, above the horizontal axis you have a semicircle with r=3, or A₁=πr²/2=9π/2.

Below the horizontal axis you have two lobes for r=3+3sin(θ), or A₂=2∫(1/2)r²dθ integrating from 3π/2 to 2π.

A = A₁+A₂

a = 3π/2, b=2π

A₂=∫_a^b(3+3sin(θ))²dθ = ∫_a^b 9 +18sin(θ)+9sin²(θ) dθ = ∫_a^b 9 +18sin(θ)+9(1-cos(2θ))/2 dθ

Which should be easy to integrate