Polar Equation Question

Question

The figure above shows the graph of the polar curve r=1−2cosθ for 0≤θ≤π and the unit circle r=1.

(a)  Find the area of the shaded region in the figure.

Question 2

(b)  Find the slope of the line tangent to the polar curve r=1−2cosθ at the point where x=−2. Show the computations that lead to your answer.

Question 3

(c)  A particle moves along the polar curve r=1−2cosθ so that dθdt=2. Find the value of drdt at θ=2π3, and interpret your answer in terms of the motion of the particle.

Answer 1

I can help you with this...but you will need to do some work yourself.

The area involved needs to be split into 2 pieces

The integrand in one piece is .(1-2 cos θ)² and the limits are θ=0 to π/2....and the result is multiplied by .5

In the other piece the integrand is 1 and the limits are π/2 to π (notice that the integrand is the constant 1) and the result is multiplied by .5

To get the slope of the tangent line:

x=(1-2 cos θ)cos θ and y=(1-2 cos θ)sin θ

Differentiate x and y with respect to θ then divide dy/dθ by dx/dθ to get dy/dx and evaluate at the point given.

In the third part differentiate with respect to t, i.e. get an expression for dr/dt as a function of dθ/dt...then substitute the given value of θ to get dr/dt.