To find the exact length of the polar curve given by r = e^(2θ) over the interval 0 ≤ θ ≤ 2π, we can use the formula for the length of a polar curve:
L = ∫ from a to b √( (dr/dθ)^2 + r^2 ) dθ
1. First, we need to calculate dr/dθ. For r = e^(2θ), we differentiate:
dr/dθ = 2e^(2θ)
2. Next, we substitute r and dr/dθ into the length formula:
L = ∫ from 0 to 2π √( (2e^(2θ))^2 + (e^(2θ))^2 ) dθ
3. Simplifying inside the square root:
√( (4e^(4θ)) + (e^(4θ)) ) = √(5e^(4θ)) = √5 * e^(2θ)
4. Now the integral becomes:
L = ∫ from 0 to 2π √5 * e^(2θ) dθ
5. Factor out √5:
L = √5 * ∫ from 0 to 2π e^(2θ) dθ
6. The integral of e^(2θ) is given by:
∫ e^(2θ) dθ = (1/2)e^(2θ)
7. Evaluating the definite integral:
∫ from 0 to 2π e^(2θ) dθ = (1/2)e^(2(2π)) - (1/2)e^(2(0)) = (1/2)e^(4π) - (1/2)
8. Substitute this result back into the equation for L:
L = √5 * [(1/2)e^(4π) - (1/2)]
9. Finally, simplifying that yields:
L = (√5/2)(e^(4π) - 1)
Thus, the exact length of the polar curve r = e^(2θ) from 0 to 2π is:
L = (√5/2)(e^(4π) – 1)