Find the area delimited by two polar curves

In summary, the conversation discusses the attempt to solve a problem involving finding the angles and area of an intersection point of two circles. The individual initially attempted to solve it using integrals, but after being told by their teacher that their solution was incorrect, they are now struggling to find an alternative method to solve the problem. The conversation also mentions the possibility of using geometry or converting to Cartesian coordinates, but questions the need for such methods. Ultimately, the teacher's rejection of the initial solution is deemed as incorrect and the individual's approach is praised.
  • #1
Astrowolf_13
1
1
Homework Statement
Find the area limited by polar curves ##r=sin(\theta)## and ##r=\sqrt 3*cos(\theta)##.
Relevant Equations
##r=sqrt(3)*cos(\theta)##, ##r=sin(\theta)##, ##0 \leq \theta \leq pi/2##.
I attempted to solve this problem by finding the angles of an intersection point by equalling both ##r=sin(\theta)## and ##r=\sqrt 3*cos(\theta)##. The angle of the first intersection point is pi/3. The second intersection point is, obviously, at the pole point (if theta=0 for the sine curve and theta=pi/2 for the cosine curve). I then attempted to find the required area by summing its "parts":
$$Area=0.5*\int_0^\frac{\pi}{3} (sin(\theta))^2\, d(\theta) + 0.5*\int_\frac{\pi}{3}^\frac{\pi}{2} (\sqrt 3 *cos(\theta))^2\, d(\theta)$$
The final area I got is ##5*\pi/24 + \sqrt 3 /4##, which I checked on the Internet. However, my teacher disregarded this solution as incorrect (because, as I quote, line at angle pi/3 is not a boundary of either ##r=sin(\theta)## or ##r=\sqrt 3 *cos(\theta)##) and asked to use a different method to find the area, and this is what I'm struggling to do.
So the question is: is there any other method of finding this area or did I get the description of the homework statement wrong?
The graph should look like this:
zQoUzGofTrWXsxrK7PgG_Capture.jpg
 
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  • #2
I like your approach.

You can solve this without integrals, however. Let O be the origin and P the other intersection of the circles. Let M be the center of the cosine circle. Draw this triangle, then find the angle phi at M. From there you can calculate the "left" part of your area with simple geometry. Do the same for the other circle.

It's also possible to convert the circles to Cartesian coordinates and integrate there, but why would someone want to do that.
 
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  • #3
Astrowolf_13 said:
However, my teacher disregarded this solution as incorrect (because, as I quote, line at angle pi/3 is not a boundary of either ##r=sin(\theta)## or ##r=\sqrt 3 *cos(\theta)##)
The graph should look like this:
View attachment 262079
Your teacher's statement that the dashed line is not a boundary of either circle is beyond ridiculous. You have a region that contains part of each circle and the dashed line is a boundary of each subarea. You have done the problem absolutely correctly. Nicely done. Your teacher deserves a demerit.
 
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Related to Find the area delimited by two polar curves

1. What is the formula for finding the area delimited by two polar curves?

The formula for finding the area delimited by two polar curves is ∫(1/2)r²dθ, where r is the distance from the origin to the curve and θ is the angle between the initial ray and the curve.

2. How do I determine the limits of integration for finding the area?

The limits of integration can be determined by finding the points of intersection between the two polar curves. These points will serve as the starting and ending angles for the integral.

3. Can I use the same formula for finding the area between any two polar curves?

Yes, the formula for finding the area between two polar curves is applicable to any two curves, as long as the limits of integration are correctly determined.

4. Do I need to convert the polar equations into rectangular form before finding the area?

No, it is not necessary to convert the polar equations into rectangular form. The formula for finding the area between two polar curves can be directly applied to the polar equations.

5. Are there any special cases I should be aware of when finding the area delimited by two polar curves?

One special case to be aware of is when the polar curves intersect at the origin. In this case, the area between the curves will be 0, as the curves will overlap at the origin.

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