Arc Length of Space Curves

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.