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The length of a curve in calculus, also known as the arc length, is a measurement of the distance along a continuous curve between two points. To calculate this, we often use an integral. The formula to determine the arc length of the function \(f(x)\) on the interval from \(x = a\) to \(x = b\) is: \[ L = \int_a^b \sqrt{1 + [f'(x)]^2}dx \] Here's a breakdown of what each part of the formula represents:

• \(\sqrt{1 + [f'(x)]^2}\): This term computes the infinitesimal arc length by considering both the vertical and horizontal change as \(x\) changes slightly along the interval.
• \(f'(x)\): This is the derivative of the function \(f(x)\), representing the slope or rate of change at each point within the interval.
• \(\int_a^b\): This is the definite integral symbol, which calculates the total arc length over the entire interval from \(a\) to \(b\).

The process involves calculating the derivative, inputting it into the formula, and evaluating the integral to find the total length of the curve. For the function \(f(x) = 2x\) over \([0, 1]\), substituting and integrating yields the arc length as \(\sqrt{5}\).

The centroid of a curve can be visualized as its center of mass, providing a balance point based on both shape and density. It's particularly useful in engineering and physics for understanding how objects weigh distributedly. Clinically, the formulas for finding the centroid \((x_c, y_c)\) of a curve are: \[ x_c = \frac{\int_a^b x \cdot \sqrt{1 + [f'(x)]^2} dx}{L} \] \[ y_c = \frac{1}{2}\frac{\int_a^b f(x) \cdot \sqrt{1 + [f'(x)]^2} dx}{L} \] These formulas take into account both the curve's shape and its arc length \(L\). Here's what to consider:

• \(x \cdot \sqrt{1 + [f'(x)]^2}\) and \(f(x) \cdot \sqrt{1 + [f'(x)]^2}\): These expressions incorporate the curve's geometry in weighing the moment contributions in horizontal and vertical directions.
• \(L\): This is the previously calculated arc length, ensuring the centroid reflects the entire curve, not just a segmented portion.

By evaluating these integrals, you can find \(x_c\) and \(y_c\), which tell you the curve's balance points horizontally and vertically. In this example, for \(f(x) = 2x\), the centroid is located at \(\left(\frac{\sqrt{5}}{3}, \frac{2\sqrt{5}}{3}\right)\).

The surface area of revolution refers to the surface area of a three-dimensional object formed by rotating a curve around an axis. It's often used to solve practical problems in engineering and physics, such as fluid dynamics and material properties. The formula for finding a surface area formed by revolving a curve \(f(x)\) around the \(x\)-axis is: \[ A = 2\pi \int_a^b f(x) \cdot \sqrt{1 + [f'(x)]^2} dx \] Key elements to understand this formula include:

• \(2\pi\): This constant arises from the circular nature of revolution—the result of spinning a curve around the \(x\)-axis.
• \(f(x) \cdot \sqrt{1 + [f'(x)]^2}\): It combines both the height of the function \(f(x)\) and the rate of change \(f'(x)\), accounting for the distance each point moves during the spin.
• \(\int_a^b\): The integral computes the sum of the infinitesimal surface elements created by the curve's rotation over the interval \([a, b]\).

In our example with \(f(x) = 2x\) and the interval \([0, 1]\), this process results in a surface area of \(4\pi\sqrt{5}\), showcasing how the curve's rotation shapes a solid surface area.