Calculus 3: Arc length and curvature

Calculate the length of the curve over the interval 1 to 4 for the vector-valued function \ln(t), 2t, t^2.

Given a curve defined by a vector-valued function $R(T)$ where $T$ varies between $a$ and $b$, find the arclength of the curve.

Given the vector function $\mathbf{R}(t) = \langle 2t \mathbf{i} + e^t \mathbf{j} + e^{-t} \mathbf{k} \rangle$, find the arc length over the interval $[0, 1]$.

Find the arc length of the vector-valued function $\mathbf{R}(t) = 3t\mathbf{i} - t\mathbf{j}$ over the interval \([0, 3]\).