Finding the Area Inside Polar Equations

For y = 2sin(3x), the period is 2pi/3, so it's pretty easy to find an interval on the x-axis that corresponds to one loop of the polar curve. Note that you don't want any y values that are negative, since your polar curve involves r^2.In summary, to find the area of the region described by the polar curves r^2=4cos(2theta) and r^2=2sin(3theta), one can sketch a graph of the equivalent Cartesian functions and find an interval on the x-axis that corresponds to one loop of the polar curve. This can provide insight into the polar curves and help determine the integration bounds for finding the area using the
  • #1
apphysicsgirl
10
0
1. Find the area of the region described:
a) inside one loop of the lemniscate r^2=4cos(2theta)
b) inside the six-petaled rose r^2=2sin(3theta)
2. A=integral [1/2 r^2 dtheta]
Are there any easy ways to determine the integration bounds? (without graphing)
Our textbook doesn't give any examples like this.
 
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  • #2
I don't think there's an easy way that doesn't involve graphing something. For the first one, sketch a graph of y = 4cos(2x) and for the second one, sketch a graph of y = 2sin(3x).

These are not polar graphs, but they can give you some insight into what the corresponding polar graphs look like.

For y = 4cos(2x), the period is pi, so it's pretty easy to find an interval on the x-axis that corresponds to one loop of the polar curve. Note that you don't want any y values that are negative, since your polar curve involves r^2.
 

FAQ: Finding the Area Inside Polar Equations

1. What is the definition of "area inside polar equations"?

The area inside polar equations refers to the area enclosed by a polar curve or a series of polar curves on a polar coordinate system. It is typically measured in square units.

2. How do you find the area inside a polar curve?

To find the area inside a polar curve, you can use the formula A = 1/2 ∫θ1θ2 r²(θ) dθ, where θ1 and θ2 are the starting and ending angles of the curve, and r(θ) is the polar equation of the curve.

3. Are there any special cases when finding the area inside polar equations?

Yes, there are a few special cases when finding the area inside polar equations. If the polar curve crosses the origin, the area needs to be divided into two parts and calculated separately. Additionally, if the polar curve is symmetric about the origin, you can simplify the calculation by only finding the area for one half and multiplying it by 2.

4. Can the area inside polar equations be negative?

No, the area inside polar equations cannot be negative. Since area is a measure of space, it is always a positive value. If the formula for finding the area results in a negative value, it typically means that the starting and ending angles were inputted in the wrong order.

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

The formula for finding the area inside polar equations involves integration, which is a concept in calculus. The process of finding the area under a curve in a polar coordinate system is similar to finding the area under a curve in a Cartesian coordinate system using the integral calculus method.

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