Finding Area in Polar Coordinates

In summary, the problem is to find the area of r = 4cos3θ in polar coordinates. The formula for finding the area is ∫ ½r²dθ. After substituting the given equation, the integral becomes ∫ (4cos3θ)²dθ, which simplifies to ½ ∫ (16cos²9θ)dθ. However, the author has made a mistake in the integral and the correct answer should be 72π. It is also important to remember that cos and sin squared have an average value of 1/2 over a period of 2pi and to not square 3θ when squaring the cosine function. It is unclear if the author is supposed to integrate
  • #1
Kawrae
46
0
The problem is to find the area of r = 4cos3θ.

I know the formula for finding the area in polar coordinates is ∫ (from α to β) ½r²dθ.

I substituted into this formula the given equation and got:
A = ½ ∫ (from 0 to 2π) (4cos3θ)²dθ
= ½ ∫ (from 0 to 2π) (16cos²9θ)dθ
= 8 [(9/2)θ + (9/4)sin18θ) |(2π - 0)]
= 8 (9π + 0)
= 72π

This answer seems really high and the answer in the books gives an answer of 4π... can someone help show me where exactly I am messing up? Thanks :)
 
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  • #2
Something nice to remember is that cos and sin squared have an average value of 1/2 over a period of 2pi
 
  • #3
Do not square 3*theta when you square the cosine function! Also you have gotten the integral (line 3 above) wrong, although it won't change your answer. Please look up the integral of cos^2 again. Finally, are you sure you are supposed to integrate from 0 to 2pi?
 

Related to Finding Area in Polar Coordinates

What is the formula for finding the area in polar coordinates?

The formula for finding the area in polar coordinates is A = 1/2 * ∫[a,b] r^2 dθ, where r represents the distance from the origin and θ represents the angle.

How do you determine the limits of integration for finding the area in polar coordinates?

The limits of integration for finding the area in polar coordinates are determined by the angle at which the curve intersects the positive x-axis. This angle will be the lower limit of integration, while the upper limit will be the angle at which the curve intersects the positive x-axis again.

Can you use the same formula to find the area for all types of polar curves?

No, the formula for finding the area in polar coordinates only works for certain types of polar curves, such as circles, cardioids, and limaçons. Other types of polar curves may require different formulas or methods for finding their area.

Do you have to convert polar coordinates to Cartesian coordinates before finding the area?

No, you do not have to convert polar coordinates to Cartesian coordinates before finding the area. The formula for finding the area in polar coordinates is specifically designed to work with polar coordinates.

What is the importance of finding the area in polar coordinates?

Finding the area in polar coordinates is important in many real-world applications, particularly in physics and engineering. It allows for the calculation of areas in non-Cartesian systems, making it a useful tool in various fields of study.

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