Question

What is the area?

Answer 1

There is no real solution to this question.

Explanation:

The integral is not valid, if you look at what happens when theta approaches 0, you will see the function goes large, thus the area between [0,1] is kind of infinity.
---by @miles-a

I'm sorry. I don't know the steps of solving this. But you can try the formula mentioned by @alvin-l-2 as the following.

Answer 2

The calculation is hard. I used http://www.wolframalpha.com/ and obtained the result(about `0.50754`).

Explanation:

This is the graph of `r=1/(1-cos3theta)` (by Wolframalpha).

Then, how to interpret the graph?
[1] The graph crosses `x=0` at `theta=pi/2`. There,
`r=1/(1-cos((3pi)/2))=1`.

[2] When does the graph cross `x=1`?
To solve this, apply the formula `x=rcostheta`.
`1=rcostheta`
`r=1/costheta`
`1/(1-cos3theta)=1/costheta`
`1-cos3theta=costheta`
`cos3theta+costheta=1`

Use the formula `cos3theta=4cos^3theta-3costheta`.
`4cos^3theta-2costheta-1=0`

The real solution for this is:
`costheta≒0.88465`
`theta≒0.48505` [rad] `≒27.79` [deg]

In cartesian form, the graph crosses `x=0` at `(0,1)` and crosses `x=1` at `(1,tan(0.48505))=(1,0.52705)`.

Now we are ready to calculate the area.
The area bounded by polar curves is `S=1/2int_alpha^beta{f(theta)}^2d` `theta`.
In this problem,
`S=1/2int_0.48505^(pi/2)(1/(1-cos3theta))^2d` `theta=0.24401`.

However, this integration is done in polar coordination. You must add the area of the right triangle with vertices(in cartesian form) `(0,0)`, `(1,0)` and `(1,0.52705)`.

The answer is `0.24401+1/2*1*0.52705=0.50754`.

By the way, calculating areas bounded by polar curves is so difficult. I don't think it is necessary for us to do these types of calculation without a calculator.

I posted a question https://socratic.org/questions/calculating-areas-bounded-by-polar-curves-looks-extremely-difficult-do-americans two months ago but nobody has answered this. What are your opinions?