91Ó°ÊÓ

To find the arc length of a curve, we first need the derivative of the function. Given the function \( f(x) = \sqrt{4-x^2} \), which represents a semicircle's upper half, we apply the chain rule for differentiation. - **Chain Rule**: This rule is vital for finding derivatives of composite functions. If you have a function of the form \( y = g(h(x)) \), the derivative \( \frac{dy}{dx} \) is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function. - For our function, we consider \( y = g(x) = \sqrt{4-x^2} \), which we can set as \( (4-x^2)^{1/2} \). 1. Differentiate the outer function: Derivative of \( z^{1/2} \) is \( \frac{1}{2} z^{-1/2} \). 2. Differentiate the inner function \( h(x) = 4 - x^2 \): Derivative is \( -2x \). - Plug these into the Chain Rule: \( \frac{dy}{dx} = \frac{1}{2} (4-x^2)^{-1/2} (-2x) = \frac{-x}{\sqrt{4-x^2}} \). This derivative is crucial for use in the arc length formula later on.

The arc length formula is our main tool for finding the exact length of a curve between two points. It's a key concept in calculus. - The general formula for the arc length \( L \) of a curve from \( x=a \) to \( x=b \) in terms of \( x \) is \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]. - This equation accounts for both the "horizontal" distance and the "vertical" distance of the curve, giving us the true, stretched length along the curve.- Simplifying the calculation is key: in our example, we found \( \left( \frac{dy}{dx} \right)^2 = \frac{x^2}{4-x^2} \).- This simplifies our formula inside the square root to a more straightforward expression: \( \sqrt{1 + \frac{x^2}{4-x^2}} = \sqrt{\frac{4}{4-x^2}} \).Thus, our integral becomes easier to solve, which is essential for an efficient calculation of arc length.

Calculating the arc length ultimately leads us to evaluate a definite integral. Definite integrals help in finding the total accumulation of quantities which, in this case, is the length of the curve.- In calculus, a **definite integral** is denoted as \( \int_{a}^{b} f(x) \, dx \), indicating the accumulation of area under a curve \( f(x) \) from \( x=a \) to \( x=b \).- For finding arc length, once we have the simplified form \( L = \int_{0}^{2} \frac{2}{\sqrt{4-x^2}} \, dx \), we recognize its shape matches that of a known integral of trigonometric function.- The result involves finding the **antiderivative** or reverse process of differentiation: 1. We identify \( \frac{2}{\sqrt{4-x^2}} \) resembling arcsine's derivative, leading to the function \( \arcsin \left( \frac{x}{2} \right) \). 2. Evaluating this from 0 to 2 gives \( \pi \), as \( 2 \cdot \left[ \arcsin(1) - \arcsin(0) \right] = 2 \cdot \frac{\pi}{2} = \pi \).Through this careful evaluation, we find the length of the arc to be precisely \( \pi \), offering a neat and clear solution to the problem.