Arc Length of Parametric Curve2

Find the arc length of the parametric curve given by $x(t) = 9t^2$ and $y(t) = 9t - 3t^3$ for $t$ in the interval $[0, 2]$.

In this problem, we are tasked with finding the arc length of a parametric curve defined by two functions of a parameter t: $x(t) = 9t^2$ and $y(t) = 9t - 3t^3$, over the interval from $t = 0$ to $t = 2$. This type of problem involves the concept of parametric equations, which are used to represent curves in the plane using a third variable, typically denoted as t. By defining curves in terms of a parameter, we can explore their geometry and structure in a flexible and often more insightful way than using traditional Cartesian coordinates.

To find the arc length of a parametric curve, you use the formula that integrates the square root of the sum of the squares of the derivatives of the x and y components with respect to t. This involves calculating $dx/dt$ and $dy/dt$, squaring these derivatives, summing them, taking the square root, and then integrating with respect to t over the given interval. This process effectively measures the "distance" along the curve as it is traced out by the parameter. Solving such problems strengthens your understanding of integration techniques and the use of derivatives in practical applications.

This topic falls under the broader subject of calculus and specifically explores the calculation of real-world quantities through parametric equations, providing a bridge between abstract mathematics and tangible problem-solving. Understanding these concepts is crucial in fields such as physics and engineering, where the path and length of curves can model real-world phenomena, from the trajectory of projectiles to the form of roller coaster tracks.

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