Arc Length in Parametric Form An arc length is the length of the curved portion of a parametric curve, which is a function that describes a path in the p...
An arc length is the length of the curved portion of a parametric curve, which is a function that describes a path in the plane. In other words, it is the total distance traveled along the path.
In parametric form, an arc length is given by the integral of the magnitude of the derivative of the parametric equation with respect to time.
In other words:
where:
s is the arc length
a and b are the start and end times of the curve
dx/dt is the derivative of the parametric equation with respect to time
For example, consider the parametric equation x = t and y = t^2. The derivative of this equation is dx/dt = 1 and dy/dt = 2t. Therefore, the arc length from t = 0 to t = 1 is:
which is the length of the entire curve.
This concept extends to higher dimensions, where the arc length is calculated by integrating the magnitude of the derivative of the parametric equation with respect to a specific direction in the higher-dimensional space