Calculus

Polar Equation 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.

Can’t post diagram because it is a url

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  1. If you can't post the diagram, we can't post answers.

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  2. (a) recall that A = ∫ 1/2 r^2 dθ

    (b) dy/dx = (dy/dθ) / (dx/dθ)
    = (r' sinθ + r cosθ)/(r'cosθ - r sinθ)
    so find r and θ when x=2, and plug and chug.

    (c) dr/dt = 2sinθ dθ/dt = 2(√3/2)(2)

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  3. the diagram is 1-2costheta graphed from 0 to pi and a circle with a radius of 1. The shaded area is almost 3/4ths the unit circle except for the slivers that 1-2costheta cuts thru the first and third quadrants

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