How do I properly bound the area between polar curves?

In summary, the individual discussing their trouble with finding the area between two polar curves has discovered that the two graphs do not intersect where they appear to and have provided a solution involving two separate integrals with different limits. They also received confirmation that their method was correct.
  • #1
itzela
34
0
I am having trouble finding the area between 2 polar curves... I have the procedure down, but the bounds are throwing me off. Any help with understanding how to bound would be great appreciated!

I have attatched one problem that I am having hard time with and the work I have done. I know that I am doing something wrong because I am getting a negative number for an area (which shouldn't be).
 

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  • #2
At Pi/2 your lower limit changes to 0. (Otherwise
you're integrating from the bottom half of the circle
up to the 1 + cos(theta) curve.)
 
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  • #3
That was tricky, but I figured out what the problem is. The two graphs don't intersect where they appear to intersect. They do intersect at [itex]\frac {\pi} {3}[/tex], but they don't really intersect at the origin. There [itex]\theta = \pi [/tex] for [itex] r=1+cos(\theta)[/tex], but [itex]\theta= \frac {\pi} {2}[/tex] for [itex] r=3cos(\theta)[/tex]. What you need to do is two separate integrals with different limits. Find the whole area for [itex] r=1+cos(\theta)[/tex] and then subtract out the area for [itex] r=3cos(\theta)[/tex].
 
  • #4
Find and double the area from [itex]\pi/2[/itex] to [itex]\pi[/itex] for the [itex]1+cos(\theta)[/itex] graph. Then, find the area from [itex]\pi/3[/itex] to [itex]\pi/2[/itex] for [itex][(1+cos(\theta))^2 - (3cos(\theta))^2][/itex]; double this as well. The total area is the sum of the two.
 
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  • #5
Thanks a bunch guys =)

What I did was:

1. found the area from [itex]\pi[/itex] to [itex]\pi/3[/itex] for the graph of [itex]1+cos(\theta)[/itex] -- doubled it

2. found the area from [itex]\pi/3[/itex] to [itex]\pi/2[/itex] for the graph [itex] 3cos(\theta)[/itex] -- doubled it

3. Substracted the value i got in #2 from #1... and that was my answer.

... Does it sound on the right path?
 
  • #6
Yup, that works.
 

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

The formula for finding the area bounded by polar curves is given by A = 1/2 ∫αβ r2(θ) dθ, where r(θ) represents the polar curve and α and β are the starting and ending angles.

2. How do you determine the starting and ending angles when finding the area bounded by polar curves?

The starting and ending angles, α and β, are determined by finding the points of intersection between the polar curves. These points represent the boundaries of the area and can be used to set the limits of integration in the formula.

3. Can you use the formula for finding the area bounded by polar curves for all types of polar curves?

No, the formula can only be used for polar curves that do not intersect themselves. If the curve intersects itself, the area needs to be split into multiple parts and the formula must be applied separately for each part.

4. How does the direction of rotation affect the calculation of the area bounded by polar curves?

The direction of rotation does not affect the calculation of the area bounded by polar curves. However, it does affect the sign of the area. Counterclockwise rotation is typically considered positive, while clockwise rotation is considered negative.

5. Is there a graphical method for finding the area bounded by polar curves?

Yes, a graphical method called the "wedge method" can be used to estimate the area bounded by polar curves. This method involves dividing the area into small sectors and approximating the area of each sector using the formula for the area of a triangle. The sum of these approximations gives an estimate of the total area.

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