Calculus with Parametric curves

Question

Find the exact length of the curve.

x = e^t − t,    y = 4e^{t / 2},    0 ≤ t ≤ 4

Answer 1

x = e^t - t

dx/dt = e^t - 1

y = 4e^t/2

dy/dt = 4e^t/2(1/2) = 2e^t/2

Length of curve = L = ∫_{(0 to 4)} √ [(dx/dt)² + (dy/dt)²]dt

L = ∫_{(0 to 4)} √ [(e^t - 1)² + (2e^t/2)²]dt = ∫_{(0 to 4)} √[e^2t - 2e^t + 1 + 4e^t] dt

L = ∫_{(0 to 4)} √ [e^2t + 2e^t + 1]dt = ∫_{(0 to 4)} √ (e^t + 1)² dt = ∫_{(0 to 4)} (e^t + 1)dt = (e^t + t)_{(0 to 4)}

L = (e⁴ + 4) - 1 = e⁴ + 3