Final answer:
To find the exact length of the polar curve r = 4 sin θ, integrate the arc length formula ∫√(r²+(dr/dθ)²)dθ using the given equation, differentiate r with respect to θ, and substitute the limits of θ.
Step-by-step explanation:
The polar curve given by r = 4 sin θ represents a cardioid. To find the length of this polar curve, we can use the arc length formula.
The formula for arc length is: Δθ = ∫ √(r² + (dr/dθ)²) dθ
In this case, r = 4 sin θ. To find (dr/dθ), we differentiate r with respect to θ.
After calculating the integral, substitute the limits of θ to find the exact length of the polar curve.