Find area of regions bounded by curve and line

[DryRun]

Homework Statement
Find area of regions bounded by
x^2 + y^2 = 9, y = 2x, x-axis in the first quadrant

The attempt at a solution
So, i drew the graph of y against x in my copybook, and circle with origin (0,0), radius = 3 units. The line y = 2x cuts through the circle.
Transforming to polar coordinates, the new limits are:
0≤θ≤tan^-1(3) and 0≤r≤3
$$\int\int rdrd\theta$$
After integration, i get the final answer: (9/2)tan^-1(3)
However, the answer in my notes is: (9/2)tan^-1(2). Did i copy the wrong answer or is my work wrong? I'm not sure.

 

[SammyS]

Homework Statement
Find area of regions bounded by
x^2 + y^2 = 9, y = 2x, x-axis in the first quadrant

The attempt at a solution
So, i drew the graph of y against x in my copybook, and circle with origin (0,0), radius = 3 units. The line y = 2x cuts through the circle.
Transforming to polar coordinates, the new limits are:
0≤θ≤tan^-1(3) and 0≤r≤3
$$\int\int rdrd\theta$$
After integration, i get the final answer: (9/2)tan^-1(3)
However, the answer in my notes is: (9/2)tan^-1(2). Did i copy the wrong answer or is my work wrong? I'm not sure.

Your answer is wrong, because the following is wrong

...the new limits are:
0≤θ≤tan^-1(3)...​

 

[DryRun]

OK, so here is the graph:
http://s2.ipicture.ru/uploads/20111227/zpUVSESp.jpg
From my understanding, i need to find the area of the red section from the graph above.

For θ fixed, 0≤θ≤cos^-1(1/√5) and 0≤r≤3
I checked from calculator and cos^-1(1/√5) = tan^-1(2)

Thanks for your help, SammyS.