Problem 24 - Calculus: Early Transcendentals

Question

Set up the integral to compute the arc length of the function on the given interval. Do not evaluate the integral. \(f(x)=\sqrt{1-x^{2}}\) on \([-1,1] .\) (Note this describes the top half of a circle with radius \(1 .)\)

Step-by-Step Solution

Answer

The integral for the arc length is \(\int_{-1}^{1} 1 \, dx\).

Step 1: Identify the Formula for Arc Length

The formula for the arc length of a curve defined by a function \(y = f(x)\) over an interval \([a, b]\) is given by: \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \] This will be used to find the arc length of the given function.

Step 2: Compute the Derivative of the Function

Given \(f(x) = \sqrt{1-x^2}\), we need to find the derivative \(\frac{df}{dx}\). Using the chain rule, which states that if \(y = g(h(x))\), then \(\frac{dy}{dx} = g'(h(x)) \cdot h'(x)\), we have: \[ \frac{d}{dx}(1-x^2)^{1/2} = \frac{1}{2}(1-x^2)^{-1/2} \cdot (-2x) = -\frac{x}{\sqrt{1-x^2}} \] So, \(\frac{df}{dx} = -\frac{x}{\sqrt{1-x^2}}\).

Step 3: Substitute the Derivative into the Arc Length Formula

Now substitute \(\frac{df}{dx} = -\frac{x}{\sqrt{1-x^2}}\) into the arc length formula. \[ \sqrt{1 + \left( \frac{df}{dx}\right)^2} = \sqrt{1 + \left(-\frac{x}{\sqrt{1-x^2}}\right)^2} = \sqrt{1 + \frac{x^2}{1-x^2}} \] Simplify this expression: \[ \sqrt{1-x^2 + x^2} = \sqrt{1} = 1 \] When simplified, the expression under the square root equals 1.

Step 4: Set Up the Integral

Substitute the simplified expression back into the integral for the arc length over the interval \([-1, 1]\): \[ L = \int_{-1}^{1} 1 \, dx \] This is the integral which represents the arc length of the function \(f(x) = \sqrt{1-x^2}\) on the interval \([-1, 1]\).

Key Concepts

Integral Calculus

Integral calculus is a core part of calculus that deals with integrals and their properties. When solving problems related to arc length, integrals help determine the length of a curve over a specified interval. In simple terms, an integral can be thought of as a sum that calculates total behavior over a continuous stretch. To find the arc length using integral calculus, we rely on the arc length formula
\[L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx\]
where \

• \(a\) and \(b\) represent the endpoints of the interval.
• \(\frac{dy}{dx}\) is the derivative of the function.
• The expression under the square root accounts for the curve's slope.

We set up the integral over the interval \([-1, 1]\) for the given function. Although the problem does not require us to evaluate, understanding the setup helps in visualizing how arc length unfolds along a curve.

Derivative Calculation

Derivatives are essential in calculus as they represent the rate of change or slope of a function at a particular point. To find the arc length of a curve, we first need the derivative \(\frac{df}{dx}\). For our function \(f(x) = \sqrt{1-x^2}\), applying the concept of derivatives calculates how quickly the function changes.
Using the chain rule, we determine the derivative as follows:

• Recognize that \(f(x) = (1-x^2)^{1/2}\).
• Apply the chain rule: \(\frac{d}{dx}(u^n) = nu^{n-1} \cdot \frac{du}{dx}\).
• Simplify the derivative to \(-\frac{x}{\sqrt{1-x^2}}\).

This derivative \(\frac{df}{dx}\) reflects changes along the curve \(f(x)=\sqrt{1-x^2}\) and plays a critical role in the arc length formula by describing the curve's slope behavior on the interval.

Chain Rule in Calculus

The chain rule in calculus is a powerful tool for differentiating composite functions, functions that can be thought of as a function within another function. It's essential to use the chain rule whenever we encounter a situation where a function depends on another function for its input.
For the given function \(f(x) = \sqrt{1-x^2}\), we recognize it as a composition of two simpler functions:

• Inner function: \(h(x) = 1 - x^2\)
• Outer function: \(g(u) = u^{1/2}\)

The chain rule states that if \(y = g(h(x))\), then the derivative \(\frac{dy}{dx}\) is \(g'(h(x)) \cdot h'(x)\), implying that each function must be differentiated and multiplied together.
By applying this process, we derived \(\frac{df}{dx} = -\frac{x}{\sqrt{1-x^2}}\). Understanding and applying the chain rule aids in navigating more complex derivative calculations and simplifies differentiating functions that might initially seem challenging.