Question

(5) a) Sketch r = 3+ 3 cosθ and b) Find the are length of the curve for 2π/3 ≤ θ ≤ π

Answer 1

As you've asked to solve the part (b), therefore, I'm solving that part.

Solution: The arc length of a polar curve r = f(θ) between θ = a and θ = b is given by the integral:

(1)

Given the curve:

  (2)

Differentiating this equation w.r.t , we'll get:

(3)

Since we've to calculate the length of the polar between,

(4)

in equation (1). Thus, we've to calculate,

Using (1), (2), (3) and (4), we'll get:

On simplifying, we'll get:

Now solving,

Rewriting using trigonometric identity,

We'll get:

On integration, we'll get:

Therefore,

Substituting limits, we'll get:

I hope it helps you!