To find the exact length of a polar curve given by the equation r = e^(3θ), we can use the formula for the length of a polar curve:
L = ∫αβ √(r² + (dr/dθ)²) dθ
In our case, we have:
r = e^(3θ)dr/dθ = 3e^(3θ)
First, we need to square r and the derivative:
r² = (e^(3θ))² = e^(6θ)(dr/dθ)² = (3e^(3θ))² = 9e^(6θ)
Now we add these two results together:
r² + (dr/dθ)² = e^(6θ) + 9e^(6θ) = 10e^(6θ)
Next, we need to take the square root:
√(r² + (dr/dθ)²) = √(10e^(6θ)) = √10 * e^(3θ)
Now we can plug this into our length formula:
L = ∫02π √10 * e^(3θ) dθ
Calculating the integral, we have:
L = √10 * ∫02π e^(3θ) dθ
To evaluate the integral, we find the antiderivative of e^(3θ):
∫ e^(3θ) dθ = (1/3)e^(3θ)
Now we can evaluate from 0 to 2π:
L = √10 * [ (1/3)e^(3(2π)) - (1/3)e^(3(0)) ]
Calculating this gives:
L = √10 * [(1/3)e^(6π) - (1/3)e^(0)] = √10 * [(1/3)e^(6π) - (1/3)]
Simplifying, we find:
L = (√10 / 3) * (e^(6π) - 1)
This gives us the exact length of the polar curve r = e^(3θ) from θ = 0 to θ = 2π.