Final answer:
The arc length of the given polar curve on the interval [0, 2] is √320 units.
Step-by-step explanation:
The given polar curve is r = 8θ2. To find the arc length on the interval [0, 2], we need to integrate the arc length formula ds = √(r^2 + (dr/dθ)^2)dθ. First, let's find the derivative dr/dθ by differentiating r = 8θ2. The derivative is dr/dθ = 16θ. Plugging this into the arc length formula, we get ds = √(64θ^2 + 256θ^2) dθ = √(320θ^2)dθ.
Integrating this, we have S = ∫√(320θ^2)dθ, where S represents the arc length. Since the interval is [0, 2], we evaluate the integral from 0 to 2: S = ∫02 √(320θ^2)dθ. Simplifying, we can factor out √320 to get S = √320 ∫02 θ dθ. Evaluating this integral, we have S = √320 [(1/2)θ^2] from 0 to 2. Plugging in the values, we get S = √320 [(1/2)(2)^2 - (1/2)(0)^2] = √320 (1 - 0) = √320.
Therefore, the arc length of the given polar curve on the interval [0, 2] is √320 units.