Area of Polar Curve: Find r = 1 + 2cos(θ)

[aub]

Homework Statement

Find the area inside the inner loop of the limacon curve : r = 1 + 2cos(θ)

Homework Equations

A = ∫$\stackrel{α}{β}$($\frac{1}{2}$r²)dθ

The Attempt at a Solution

i have the solution, my question is : how do you find α and β ?
here α = 2π/3 and β = π

A = 2∫$\stackrel{2π/3}{π}$($\frac{1}{2}$r²)dθ

 

[SammyS]

Homework Statement

Find the area inside the inner loop of the limacon curve : r = 1 + 2cos(θ)

Homework Equations

A = ∫$\stackrel{α}{β}$($\frac{1}{2}$r²)dθ

The Attempt at a Solution

i have the solution, my question is : how do you find α and β ?
here α = 2π/3 and β = π

A = 2∫$\stackrel{2π/3}{π}$($\frac{1}{2}$r²)dθ

Have done the polar graph of r = 1 + 2cos(θ)?

Have you done the Cartesian graph ?

 

[aub]

Have done the polar graph of r = 1 + 2cos(θ)?

Have you done the Cartesian graph ?

i did the polar one, i can see that its tangent to 2∏/3 on the origin but i can't get the point if there is any

and for the cartesian one, we never did that in class for those curves..


what were you hinting about?

thanks

 

[SammyS]

i did the polar one, i can see that its tangent to 2∏/3 on the origin but i can't get the point if there is any

and for the cartesian one, we never did that in class for those curves..

what were you hinting about?

thanks

You need to find the range of θ values that correspond to the inner loop, correct?

Looking at both graphs might help. Also, solving

1+2cos(θ) = 0​

for θ might help.

What is unusual about the values of r on the inner loop?

 

[aub]

You need to find the range of θ values that correspond to the inner loop, correct?

Looking at both graphs might help. Also, solving

1+2cos(θ) = 0​

for θ might help.

yeah, i thought what i need to do is solving for θ, it gave me α = 2∏/3 and β = 4∏/3 and i here i don't need to multiply the integral by 2 anymore.. I am lost

What is unusual about the values of r on the inner loop

do you mean the symmetry?

 

[aub]

also, if anyone can help me to find the area of the region R between the inner loop and outer loop
what would be α and β ?

 

[aub]

i guess i got it just tell me if its right

so for the area of the inner loop A₁, α = 2∏/3 and β = 4∏/3 and i apply the formula

for the area between the inner loop, i find the whole area with α = 0 and β = 2π then substract A₁

 

[SammyS]

What is unusual about the values of r on the inner loop?

do you mean the symmetry?

No. For $\displaystyle \frac{2\pi}{3}<\theta<\frac{4\pi}{3}\,,$ 1 + 2cos(θ) is negative, so r is negative.