Arc Length and Surface Area of Revolution

Given $x^3 = y^5 + 2$, find the arc length from $y = 1$ to $y = 3$ and the surface area when the arc is rotated about the x-axis.

This problem involves calculus concepts related to finding the arc length of a curve and the surface area of a solid of revolution. It requires understanding the relationship between the variables in the given equation, $x^3 = y^5 + 2$, and how to express the curve parametrically or in terms of one variable for ease of integration. Additionally, knowing the geometric meaning behind the equation aids in visualizing how the curve behaves between the given limits, from $y = 1$ to $y = 3$.

Calculating arc length typically involves using the arc length formula, which is an integral of the square root of 1 plus the derivative of the function squared over the desired interval. Recognizing which variable to differentiate and integrate with respect to is crucial in setting up the problem correctly.

To find the surface area of the solid formed when the arc is revolved about the x-axis, one must use the surface area formula specific for solids of revolution. This involves integrating $2\pi$ times the function (related to the radius of revolution) multiplied by the arc length differential. Both these processes lean heavily on your ability to handle integrals and understand the implications of rotating a curve around an axis.

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