Finding area between two curves Polar Coordinates

In summary, the question asks to find the area inside a circle with radius 3sinθ and outside a carotid with radius 1 + sinθ. The solution involves finding the limits of integration, graphing the two shapes and finding their points of intersection, and using the formula for polar area to find the integral of (3sinθ - (1+sinθ)). Wolfram Alpha can be used to check the final answer.
  • #1
PsychonautQQ
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Homework Statement


Find the area inside the circle r = 3sinθ and outside the carotid r = 1 + sinθ




The Attempt at a Solution


Alright so I graphed it and found that they intersect at ∏/6 and 5∏/6.
I can't think of a good way to approach the problem. The carotid has some of it's area beneath the x-axis otherwise I would take the area of the 3sinθ - the area of the other one. Will it work if set the limits of integration to pi/6 and 5pi/6 and take the integral of (3sinθ - (1+sinθ)? I'm a bit lost
 
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  • #2
PsychonautQQ said:

Homework Statement


Find the area inside the circle r = 3sinθ and outside the carotid r = 1 + sinθ




The Attempt at a Solution


Alright so I graphed it and found that they intersect at ∏/6 and 5∏/6.
I can't think of a good way to approach the problem. The carotid has some of it's area beneath the x-axis otherwise I would take the area of the 3sinθ - the area of the other one. Will it work if set the limits of integration to pi/6 and 5pi/6 and take the integral of (3sinθ - (1+sinθ)? I'm a bit lost

Yes, that all seems reasonable. Just compute your integral and you're done.
 
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  • #3
I got 1.369705. Is there any online integral doer I can use to check my work?
 
  • #4
PsychonautQQ said:
I got 1.369705. Is there any online integral doer I can use to check my work?

Yes, that's the correct answer. Wolfram alpha is good for checking your work afterwards.
 
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  • #5
PsychonautQQ said:

Homework Statement


Find the area inside the circle r = 3sinθ and outside the carotid r = 1 + sinθ




The Attempt at a Solution


Alright so I graphed it and found that they intersect at ∏/6 and 5∏/6.
I can't think of a good way to approach the problem. The carotid has some of it's area beneath the x-axis otherwise I would take the area of the 3sinθ - the area of the other one. Will it work if set the limits of integration to pi/6 and 5pi/6 and take the integral of (3sinθ - (1+sinθ)? I'm a bit lost

NO. That's the wrong integrand. Look up the formula for polar area.
 

Related to Finding area between two curves Polar Coordinates

1. What are polar coordinates and how are they different from Cartesian coordinates?

Polar coordinates are a system of representing points in a two-dimensional plane using a distance (r) and angle (θ) from a fixed point called the pole. This is different from Cartesian coordinates, which use a horizontal (x) and vertical (y) distance from a fixed point.

2. How do I find the area between two curves using polar coordinates?

To find the area between two curves using polar coordinates, you can use the formula A = 1/2 ∫(r2 - r1)^2 dθ, where r1 and r2 are the equations of the two curves and θ is the angle between them. This integral can be evaluated using standard integration techniques.

3. Can I use polar coordinates to find the area between two curves that intersect?

Yes, you can use polar coordinates to find the area between two curves that intersect. In this case, you will need to break the integral into smaller parts and evaluate each one separately, using the angles at which the curves intersect as the limits of integration.

4. Are there any limitations to using polar coordinates to find area between two curves?

One limitation of using polar coordinates is that they are only suitable for finding the area between curves that can be expressed in polar form. Additionally, the curves must be simple and not intersect or overlap too much, as this can make the integral more difficult to evaluate.

5. Can I use polar coordinates to find the area between curves in three dimensions?

No, polar coordinates are only suitable for finding the area between curves in two dimensions. In three dimensions, you would need to use a different coordinate system, such as cylindrical or spherical coordinates, to find the area between curves.

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