Find Curve with Given Length Integral Through a Point

Find a curve through the point $(2, -4)$ whose length integral from $y = 1$ to $y = 2$ is given by $\displaystyle \int_{1}^{2} \sqrt{1 + \left( \frac{4}{y^3} \right)^2} \, dy$.

The problem presented is a rich exploration into the application of calculus in determining a specific curve based on integral properties. The essence of this problem is rooted in the concept of arc length, where one needs to decipher the path of a curve given certain constraints. In this scenario, the curve must pass through a specified point while its length, as described by an integral over a given interval, satisfies a particular equation. This task is not only about calculating the length of an arc but also involves understanding how to manipulate the equation of a curve so that the integral criterion is satisfied.

A strategic approach to solve this involves recognizing the integral as an arc length integral. The standard formula for arc length introduces the square root of the sum of squares of the derivative of a function. Here, the integral is presented in terms of y, suggesting that expressing x as a function of y might simplify the integration process. Exploring parametric equations or considering transformations that simplify the integrand can prove beneficial. The mathematical insight comes from identifying the form of the derivatives and the subsequent simplification of the root expression, leading to an accessible integral.

Additionally, this problem serves as an insight into how differential calculus interacts with integral calculus. By determining a curve that satisfies these geometric and integral conditions, the problem offers a practical perspective on how calculus describes and accommodates the shape and dimensions of curves in a coordinate plane. On a broader level, such problems highlight the intricate interplay between various calculus concepts essential for tackling complex problems or modeling real-world scenarios.

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