Surface Area of Solid of Revolution

Find the surface area of the solid formed by revolving the curve $y=\sqrt{4x-x^2}$ about the x-axis, where x ranges from $0.5$ to $1.5$.

This problem involves finding the surface area of a solid of revolution, which is a common type of problem in calculus when exploring applications of integration. The surface area of a solid of revolution is determined by revolving a curve about a line, in this case, the x-axis. To tackle these problems, it's essential to choose the correct formula that accommodates the axis of rotation and the nature of the curve. In this scenario, the curve given is expressed in terms of y and revolves around the x-axis.

One must understand the fundamental relationship between the curve, its rotation, and the resulting surface geometry. You'll be using the concept of integrating along an axis to sum up infinitesimally small slices—or rings—that form the surface of the solid. This involves an integral setup that considers the derivative of the curve function to ensure the circumferential expansion of the surface is accurately captured. Recognizing the bounds for the integral, which in this problem are from $0.5$ to $1.5$, helps to confine the area of interest.

As you work through this problem, you'll rely heavily on skills related to setting up the integral with respect to the axis of rotation and ensuring the proper evaluation of that integral. These techniques emphasize the importance of understanding the geometric and analytical aspects of calculus, as well as the necessity of careful algebraic manipulation when dealing with square roots and rational functions within your calculations. The problem serves not only as practice for applying integration techniques but also as an opportunity to visualize and connect the abstract mathematical concepts with tangible geometric interpretations.

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