12.6
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’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 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 Δt. 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 Δt,
\begin{equation*}\frac{\mathbf{r}(t_i+\Delta t)-\mathbf{r}(t_i)}{\Delta t}\approx\mathbf{r}'(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}'(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}'(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}'(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}'(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*}
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’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.
From Chapter 12:
Now Playing
Vector Functions and Motion
27 Views
Vector Functions and Motion
126 Views
Vector Functions and Motion
56 Views
Vector Functions and Motion
51 Views
Vector Functions and Motion
49 Views
Vector Functions and Motion
9 Views
Vector Functions and Motion
14 Views
Vector Functions and Motion
13 Views
Vector Functions and Motion
14 Views
Vector Functions and Motion
11 Views
Vector Functions and Motion
12 Views
Vector Functions and Motion
11 Views
Vector Functions and Motion
16 Views