Arc Length and Surface Area of Rotated Curve

Given $x = -y^2 + 4y$, find the arc length from $y = 0$ to $y = 4$ and the surface area when the arc is rotated about the x-axis and y-axis.

This problem involves calculating the arc length of a curve defined by the equation $x = -y^2 + 4y$. Arc length problems often require integrating a special formula that calculates the length of a curve over a specified interval. In this case, we're looking at the interval from $y = 0$ to $y = 4$. The process involves recognizing the derivative of the function with respect to y, and integrating over the given range.

The second part of the problem introduces the concept of the surface area of revolution, which involves revolving the curve around an axis. This problem asks for the computation of surface areas when the curve is revolved both around the x-axis and the y-axis. Similar to volumes of revolution, calculating the surface area involves integrating a unique formula which takes into account the radius at each point along the curve as it revolves.

Understanding these concepts requires a strong grasp of integration and the ability to manipulate and apply the appropriate formulas. The challenge lies in setting up the integrals correctly and recognizing the geometric interpretation behind these advanced calculus operations. This type of problem is typical in calculus when discussing applications of definite integrals, specifically related to arc lengths and areas of surfaces generated by revolving curves.

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