Arc length of parametric curve23

Find the arc length of the curve given that $\vec{r}(t) = 3\cos(t)\hat{i} + 3\sin(t)\hat{j} + 6t\hat{k}$ where $t$ is from 0 to $\pi$.

When asked to find the arc length of a curve given in vector form, like in this problem, it's important to start by recalling the general formula for arc length. In three-dimensional space, the arc length of a smooth vector function can be determined through integration. Specifically, the formula involves integrating the magnitude of the derivative of the vector function over the given interval. Understanding how to differentiate the vector function with respect to the parameter is crucial here. The result is a new vector whose magnitude you integrate to find the total length of the curve over the specified interval.

It's also important to grasp the geometric interpretation of the problem. The arc length essentially measures the length of the path traced by the curve in space. As you work through such problems, consider both the algebraic manipulation to find the derivative and the conceptual understanding of what differentiating the vector function represents—in this case, it relates to the velocity of a point moving along the curve. Once the derivative is determined, calculating its magnitude at every point along the curve and summing these infinitesimal distances via integration gives the total arc length.

In terms of problem-solving strategy, evaluating integrals involving trigonometric functions, as is common with vector functions involving cosines and sines, often requires trigonometric identities or substitution to simplify. Additionally, always keep an eye on the limits of integration as they define the actual portion of the curve you are seeking to measure. This example provides a great opportunity to practice these crucial techniques in calculus and vector geometry.

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