Final answer:
To find the length of the spiraling polar curve, we can use the formula for the arc length of a polar curve.
Step-by-step explanation:
The length of a polar curve can be found using the formula:
L = ∫√r² + (dr/dθ)² dθ
In this case, the polar curve is described by r = 4e^(5θ). To find the length from 0 to 2π, we evaluate the integral:
L = ∫√(16e^(10θ) + 25e^(10θ)) dθ
By simplifying the expression within the square root and evaluating the integral, we can find the length of the spiraling polar curve.