Area between 2 polar equations

In summary, the conversation discusses finding the area of a region inside one curve and outside another curve using the equation A=1/2*r^2*theta. The suggested solution involves using the integrals of both curves from 0 to 2pi, but the correct range is actually from π to 2π. Plotting can be used to determine this range.
  • #1
Zack K
166
6

Homework Statement


Find the area of the region that lies inside the first curve and outside the second curve.
##r=6##
##r=6-6sin(\theta)##

Homework Equations


##A=\frac {1} {2}r^2\theta##

The Attempt at a Solution

\[/B]
If I'm correct, the area should just be ##\frac {1} {2}\int_{0}^{2\pi} 6^2 d\theta - \frac {1} {2}\int_{0}^{2\pi} (6-6sin(\theta))^2 d\theta##. If that's the case then I'm probably making simple error calculating it.

EDIT: I just realized by graphing that the range is between π to 2π. How would I figure that out? Would it be just by setting both equations equal and finding intercepts?
 
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  • #2
Zack K said:

Homework Statement


Find the area of the region that lies inside the first curve and outside the second curve.
##r=6##
##r=6-6sin(\theta)##

Homework Equations


##A=\frac {1} {2}r^2\theta##

The Attempt at a Solution

\[/B]
If I'm correct, the area should just be ##\frac {1} {2}\int_{0}^{2\pi} 6^2 d\theta - \frac {1} {2}\int_{0}^{2\pi} (6-6sin(\theta))^2 d\theta##. If that's the case then I'm probably making simple error calculating it.

EDIT: I just realized by graphing that the range is between π to 2π. How would I figure that out? Would it be just by setting both equations equal and finding intercepts?
Plotting is the way to go, but I don't think your proposed limits are correct. Plot a few easy points and you will see. Remember you want outside the second curve.
 

1. What is the area between two polar equations?

The area between two polar equations refers to the region enclosed by the two curves on a polar coordinate system. It can also be thought of as the area of a shape created by connecting the two curves with a line segment and the origin.

2. How do you find the area between two polar equations?

To find the area between two polar equations, you can use the formula A = 1/2 ∫(r₁² - r₂²) dθ, where r₁ and r₂ are the two polar equations and θ represents the angle of rotation. This formula is derived from the formula for finding the area of a sector of a circle.

3. Can the area between two polar equations be negative?

No, the area between two polar equations cannot be negative. The integral used to find the area always yields a positive value, representing the magnitude of the area.

4. Are there any special cases when finding the area between two polar equations?

Yes, there are two special cases to consider when finding the area between two polar equations. The first is when the two curves intersect and create a region with multiple areas. In this case, you would need to break up the integral into smaller parts. The second is when one of the curves is inside the other, in which case the area would be the difference between the two curves' areas.

5. How is the area between two polar equations related to calculus?

The area between two polar equations is related to calculus through the use of integrals. Calculus allows us to find the area of irregular shapes, such as those created by polar equations, by breaking them down into smaller, simpler parts and using integration to find the total area.

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