Find area of regions bounded by curve and line

In summary, the task is to find the area of the regions bounded by x^2 + y^2 = 9, y = 2x, and the x-axis in the first quadrant. The attempt at a solution involved drawing a graph and transforming to polar coordinates. However, the final answer given was (9/2)tan^-1(3), while the answer in the notes was (9/2)tan^-1(2). After further examination, it was found that the limits for theta were incorrect, and the correct limits are 0≤θ≤cos^-1(1/√5).
  • #1
DryRun
Gold Member
838
4
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.
 
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  • #2
sharks said:
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
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:
 

1. How do I find the area of a region bounded by a curve and line?

To find the area of a region bounded by a curve and a line, you can use the integration method. First, find the points of intersection between the curve and the line. Then, set up the integral by subtracting the equation of the line from the equation of the curve. Finally, integrate the resulting equation between the points of intersection to find the area.

2. What is the difference between finding the area using integration and using geometry?

The difference between finding the area using integration and using geometry is that integration allows you to find the area of regions with more complex shapes, while geometry is limited to basic shapes such as rectangles and triangles. Integration also provides a more accurate and precise result.

3. Can the area of a region bounded by a curve and line be negative?

No, the area of a region bounded by a curve and line cannot be negative. This is because area is a measurement of the space within a shape, and space cannot have a negative value. If the resulting area from integration is negative, it means that the bounds were incorrectly set up and need to be adjusted.

4. What is the significance of finding the area of regions bounded by curves and lines?

Finding the area of regions bounded by curves and lines is significant in many fields such as physics, engineering, and economics. It allows for the calculation of important quantities such as volume, surface area, and work. It also helps in solving optimization problems and understanding the behavior of functions.

5. Can the integration method be used to find the area of regions bounded by multiple curves?

Yes, the integration method can be used to find the area of regions bounded by multiple curves. In this case, you would need to find the points of intersection between each curve and set up multiple integrals to find the area of each individual region. Then, you can add or subtract the resulting areas to find the total area of the region bounded by the multiple curves.

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