Arc Length of a VectorValued Function

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

When tasked with finding the arc length of a vector-valued function, a key concept is understanding how to handle the components of the vector independently and then combine them to determine the length over a specified interval. The process begins by interpreting the given vector function, which in this example is represented in terms of the parameter t, for each of its i and j components.

The arc length of a vector function over an interval can be computed by integrating the magnitude of its derivative. Specifically, you first differentiate each component of the vector function with respect to t. Then, compute the magnitude of this derivative vector, which involves taking the square root of the sum of the squares of its components. Finally, you integrate this magnitude over the given interval. This integral yields the total length of the curve described by the vector function from the start to the end of the interval.

This technique not only applies to simple two-dimensional cases but extends to more complex multi-dimensional vector functions. The concept links closely with physical interpretations, such as the actual path length traveled by an object moving in space, reinforcing both abstract mathematical understanding and practical application. Arc length problems like this one solidify core ideas in calculus involving parametric curves and geometric interpretations, preparing students for more advanced studies in mathematics and physics.

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