Need Guidance: Area in between Polar Curves

In summary: That might give you a starting point for your integration. In summary, the problem asks to find the area of the region inside two circles, r = 2sin(x) and r = sin(x) + cos(x). Using the equation A = (1/2)(int from a to b): r^2 dx, the student is struggling to set up the intervals of integration. They suggest setting the r values in the two equations equal and solving for the angle as a starting point for integration.
  • #1
GavinMath
1
0

Homework Statement



Find the area of the region that lies inside both of the circles
r = 2sin(x)
r = sin(x) + cos(x)

Homework Equations



A = (1/2)(int from a to b): r^2 dx

(I apologize because I do not know how to make calculus look proper in text form)

The Attempt at a Solution



What I need is some theoretical help. Through graphing these circles I can see that they intersect at pi/4. However, I see that they intersect near the origin, however I can not get a common angle, which makes me confused on how to set up the intervals of my integration. Any ideas to get me going would be much appreciated!
 
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  • #2
GavinMath said:

Homework Statement



Find the area of the region that lies inside both of the circles
r = 2sin(x)
r = sin(x) + cos(x)

Homework Equations



A = (1/2)(int from a to b): r^2 dx

(I apologize because I do not know how to make calculus look proper in text form)

The Attempt at a Solution



What I need is some theoretical help. Through graphing these circles I can see that they intersect at pi/4. However, I see that they intersect near the origin, however I can not get a common angle, which makes me confused on how to set up the intervals of my integration. Any ideas to get me going would be much appreciated!

What about setting the r values in the two equations equal, and solving for the angle?
 

FAQ: Need Guidance: Area in between Polar Curves

What is the area in between polar curves?

The area in between polar curves refers to the region enclosed by two polar curves on a polar coordinate system. It is the area bounded by the curves and the origin.

How do you find the area in between polar curves?

To find the area in between polar curves, you can use the formula A = 1/2 ∫(r2 - r1)² dθ, where r1 and r2 are the two polar curves and θ represents the angle of rotation. This formula is similar to the formula for finding the area between two curves in Cartesian coordinates.

Can polar curves overlap?

Yes, polar curves can overlap and still create a bounded region. In this case, you will need to find the points of intersection and set up separate integrals for each section of the region.

What is the difference between finding the area between polar curves and finding the area between Cartesian curves?

The main difference is that polar curves are defined by a radius and an angle, while Cartesian curves are defined by x and y coordinates. This means that when finding the area between polar curves, you will need to use polar coordinates and the formula mentioned earlier.

Are there any special cases when finding the area between polar curves?

Yes, there are a few special cases to consider when finding the area between polar curves. These include curves that intersect at the origin, curves that overlap, and curves that have multiple points of intersection. In these cases, you may need to break up the region into smaller sections and use separate integrals for each section.

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