MHB What is the area inside the lemniscate and outside the circle?

  • Thread starter Thread starter Fernando Revilla
  • Start date Start date
  • Tags Tags
    Area
AI Thread Summary
The discussion focuses on calculating the area inside the lemniscate defined by r^2=150cos(2θ) and outside the circle r=5√3. By utilizing symmetry, the area is expressed as A=2A_1, where A_1 represents the area in the right half-plane. The intersection points are determined by setting the equations equal, leading to θ values of ±π/6. The area is then computed using the integral of the difference between the lemniscate and circle equations, resulting in A=75(√3 - π/3). This calculation provides a clear method for determining the desired area in polar coordinates.
Fernando Revilla
Gold Member
MHB
Messages
631
Reaction score
0
I quote a question from Yahoo! Answers

Find the area inside the lemniscate r^2=150cos(2theta) and outside the circle r=5sqrt(3).

I have given a link to the topic there so the OP can see my response.
 
Mathematics news on Phys.org
By symmetry arguments, the area is $A=2A_1$ where $A_1$ is the corresponding area on the right half plane. Equating modules, $$150\cos 2\theta=(4\sqrt{3})^2\Leftrightarrow 150\cos 2\theta=75 \Leftrightarrow \cos 2\theta=\frac{1}{2}\Leftrightarrow \theta=\pm \frac{\pi}{6}$$ Using a well known formula: $$A=2A_1=2\cdot\frac{1}{2}\int_{-\pi/6}^{\pi/6}\left(r_2^2-r_1^2\right)d\theta=\int_{-\pi/6}^{\pi/6}\left(150\cos 2\theta-75\right)d\theta\\=\left[75\sin 2\theta-75\theta\right]_{-\pi/6}^{\pi/6}=75\left(\sqrt{3}-\frac{\pi}{3}\right)$$
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Back
Top