길이 공식 L = \displaystyle \int_{a}^{b} \lvert r'(t) \rvert dt.

1. r'(t): \displaystyle r'(t) = \langle 1, -3\sin t, 3\cos t\rangle

2. \lvert r'(t) \rvert: \displaystyle \lvert r'(t) \rvert = \sqrt{1^{2} + (-3\sin t)^{2} + (3\cos t)^{2}} = \sqrt{1 + 9(\sin^{2} t + \cos^{2} t)} = \sqrt{10}

3. 적분: \displaystyle L = \int_{0}^{4} \sqrt{10}\, dt = [\sqrt{10}\, t]_{0}^{4} = 4\sqrt{10}