Name: distance traveled along a space path
Uploaded: 2026-06-08T09:38:48.158Z
Duration: 1 min 21 s
Channel: JoVE

Imagine a helicopter taking off and traveling along a curved path in three-dimensional space.To calculate the fuel required for the entire journey, the straight-line distance between takeoff and landing is not enough. The actual length of the curved path must be determined.This path is described by a position vector function that gives the helicopter&#x2019;s location at any moment.To understand the total distance traveled, we can examine a smaller segment of the path.Each tiny portion of the curve appears almost like a straight-line segment. By dividing the curve into many small segments and adding their lengths, we can approximate the total distance along the curve.As these segments become smaller and more numerous, the approximation becomes more accurate.By taking the limit as the number of segments approaches infinity, we get the exact length of the curve. Since each segment length represents the distance traveled during a small interval of time, its magnitude depends on the derivative of the position vector with respect to time.Integrating this magnitude over the interval gives the total arc length.

arc length of space curves

Arc length represents the total distance traveled along a curve in space. For a moving object such as a helicopter, the path can be modeled by a vector-valued position function\begin{equation*}\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle\end{equation*}where t denotes time. Unlike displacement, which measures only the straight-line distance between two points, arc length accounts for every change in direction along the trajectory.To calculate arc length, the interval of motion is divided into many small time intervals of size &#x394;t. Over each interval, the curve can be approximated by a short straight segment. The displacement during one interval is\begin{equation*}\Delta \mathbf{r}_i=\mathbf{r}(t_i+\Delta t)-\mathbf{r}(t_i)\end{equation*}Using the difference quotient,\begin{equation*}\Delta \mathbf{r}_i=\left[\frac{\mathbf{r}(t_i+\Delta t)-\mathbf{r}(t_i)}{\Delta t}\right]\Delta t\end{equation*}and for sufficiently small &#x394;t,\begin{equation*}\frac{\mathbf{r}(t_i+\Delta t)-\mathbf{r}(t_i)}{\Delta t}\approx\mathbf{r}&#x27;(t_i)\end{equation*}Therefore, the length of each small segment can be approximated by\begin{equation*}s_i \approx |\Delta \mathbf{r}_i|\approx|\mathbf{r}&#x27;(t_i)|\Delta t\end{equation*}Adding all segment lengths produces a Riemann sum approximation for the total distance traveled:\begin{equation*}S \approx \sum |\mathbf{r}&#x27;(t_i)|\Delta t\end{equation*}Taking the limit as the number of intervals approaches infinity gives the exact arc length formula for a space curve:\begin{equation*}S=\int_a^b |\mathbf{r}&#x27;(t)|\,dt\end{equation*}If\begin{equation*}\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle\end{equation*}then\begin{equation*}|\mathbf{r}&#x27;(t)|=\sqrt{\left(\jfrac{dx}{dt}\right)^2+\left(\jfrac{dy}{dt}\right)^2+\left(\jfrac{dz}{dt}\right)^2}\end{equation*}Thus, the arc length of a space curve from t=a to t=b is\begin{equation*}S=\int_a^b\sqrt{\left(\jfrac{dx}{dt}\right)^2+\left(\jfrac{dy}{dt}\right)^2+\left(\jfrac{dz}{dt}\right)^2}\,dt\end{equation*}

Arc Length of Space Curves

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JoVE Core

Calculus

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Vector Functions and Motion

KEY TERMS AND CONCEPTS

distance traveled along a space path

Arc length gives the total distance traveled along a curve in space. For a moving object such as a helicopter, the path can be written as a vector-valued position function r(t) = , where t is time. This distance is not the same as displacement, which measures only the straight-line change between two points. Arc length includes every change in direction along the route.

To find this length, the motion interval is split into many small time steps of size Δt. Over each step, the curve is treated as a short straight segment. The displacement for one step is Δr_i = r(t_i + Δt) - r(t_i). Using the difference quotient, this can also be written as [r(t_i + Δt) - r(t_i)]/Δt · Δt.

When Δt is very small, the difference quotient is close to the derivative r'(t_i). That means each small segment has length s_i ≈ |Δr_i| ≈ |r'(t_i)|Δt. Adding all of these segment lengths gives a Riemann sum approximation for the total distance traveled: S ≈ Σ |r'(t_i)|Δt. As the number of intervals grows without limit, this sum becomes the exact arc length.

The arc length of a space curve from t = a to t = b is S = ∫_a^b |r'(t)| dt. If r(t) = , then the speed term is |r'(t)| = sqrt((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2). So the final arc length formula for a space curve is S = ∫_a^b sqrt((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2) dt.

Distance Traveled Along a Space Path

introduction to vector functions

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Arc length gives the total distance traveled along a curve in space. For a moving object such as a helicopter, the path can be written as a vector-valued position function r(t) = , where t is time. This distance is not the same as displacement, which measures only the straight-line change between two points. Arc length includes every change in direction along the route. To find this length, the motion interval is split into many small time steps of size Δt. Over each step, the curve is treated...

curvature and its interpretation

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the normal and binormal vectors

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torsion in vector calculus

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projectile motion

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vector functions and motion: problem solving

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key terms and concepts

Learn how arc length measures the total distance along a space curve, using small line segments, derivatives, and the integral formula for x, y, and z paths.

Distance Traveled Along a Space Path | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Distance Traveled Along a Space Path

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