Question

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

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

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!

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