Find area of regions bounded by curve and line

  • Thread starter DryRun
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  • #1
DryRun
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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
[tex]\int\int rdrd\theta[/tex]
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.
 

Answers and Replies

  • #2
SammyS
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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
[tex]\int\int rdrd\theta[/tex]
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)...​
 
  • #3
DryRun
Gold Member
838
4
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.:smile:
 

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