Problem 21 - Single Variable Calculus

Question

Write an integral giving the arc length of the graph of the equation from \(P\) to \(Q\) or over the indicated interval. $$ y=\frac{1}{x^{2}+1} ; \quad P\left(-1, \frac{1}{2}\right), Q\left(2, \frac{1}{5}\right) $$

Step-by-Step Solution

Answer

The integral representing the arc length of the graph of the equation \(y=\frac{1}{x^2+1}\) between points P(-1, 1/2) and Q(2, 1/5) is: \[L=\int_{-1}^2 \sqrt{1+\frac{4x^2}{(x^2+1)^4}} dx\]

Step 1: Find the derivative of the given function

To find the derivative, we will use the power rule: \[\frac{dy}{dx} = \frac{d}{dx}\left(\frac{1}{x^2+1}\right)\] Now, rewrite the function in a form that allows us to use the power rule: \[y = (x^2 + 1)^{-1}\] Now, apply the power rule: \[\frac{dy}{dx} = -1(x^2 + 1)^{-2}(2x)\] Simplify: \[\frac{dy}{dx} = \frac{-2x}{(x^2 + 1)^2}\]

Step 2: Substitute the derivative in the arc length formula

Replace \(f'(x)\) in the arc length formula with the derivative we found in Step 1: \[L=\int_a^b \sqrt{1+\left(\frac{-2x}{(x^2+1)^2}\right)^2} dx\] Simplify the expression inside the square root: \[L=\int_a^b \sqrt{1+\frac{4x^2}{(x^2+1)^4}} dx\]

Step 3: Determine the limits of integration

The limits of integration correspond to the x-coordinates of points P and Q. For point P(-1, 1/2), the x-coordinate is -1, which corresponds to the lower limit of the integral. For point Q(2, 1/5), the x-coordinate is 2, which corresponds to the upper limit. Therefore, the integral becomes: \[L=\int_{-1}^2 \sqrt{1+\frac{4x^2}{(x^2+1)^4}} dx\]

Step 4: Write the final integral giving the arc length

The integral representing the arc length of the graph of the equation \(y=\frac{1}{x^2+1}\) between points P(-1, 1/2) and Q(2, 1/5) is: \[L=\int_{-1}^2 \sqrt{1+\frac{4x^2}{(x^2+1)^4}} dx\]

Key Concepts

Arc Length

Arc length is a key concept in Integral Calculus that helps us determine the distance from one point to another along a curve. When dealing with the graph of a function between two points, calculating the arc length means finding the total length of the curve between those points.
To find the arc length, we typically use the formula:

• \[L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx\]

Here, \( a \) and \( b \) are the limits of integration, representing the x-coordinates of the points from which we measure the arc length.
In this context, understanding arc length involves:

• Determining the derivative \( \frac{dy}{dx} \) of the function.
• Plugging this derivative into the arc length formula.
• Evaluating the integral between the given limits.

Derivative

A derivative represents the rate at which a function is changing at any given point and is a fundamental idea in Calculus. In our scenario, we deal with the function \( y = \frac{1}{x^2 + 1} \).
To find its derivative, we apply rules such as the power and chain rules.

• Start by rewriting the function to a more computation-friendly form: \( y = (x^2 + 1)^{-1} \).
• Using the power rule \( \frac{d}{dx}[u^n] = nu^{n-1}\frac{du}{dx} \), compute the derivative: \[ \frac{dy}{dx} = -1(x^2 + 1)^{-2}(2x) \]
• After simplifying, this yields \( \frac{dy}{dx} = \frac{-2x}{(x^2 + 1)^2} \).

This derivative is crucial when substituting back into the arc length formula to find the desired measurement.

Limits of Integration

Limits of integration define the interval over which we evaluate an integral. In arc length problems, these limits correspond to the x-coordinates of the points you're interested in.
For this exercise, we want to find the arc length of the function \( y = \frac{1}{x^2 + 1} \), from point \( P(-1, \frac{1}{2}) \) to point \( Q(2, \frac{1}{5}) \).
Here are the steps:

• Identify the x-coordinates of points \( P \) and \( Q \): \( -1 \) and \( 2 \) respectively.
• Use these x-coordinates as your limits of integration in the arc length integral: \[ \int_{-1}^2 \sqrt{1 + \left( \frac{-2x}{(x^2+1)^2} \right)^2} \, dx \]

By setting correct limits of integration, students can ensure they are measuring the right section of the curve.

Power Rule

The power rule is an essential tool for finding derivatives and is key to efficiently solving problems in calculus. It states that for any function \( y = x^n \), the derivative \( \frac{dy}{dx} = nx^{n-1} \).
When using the power rule in different forms:

• If the function is in a simple power form like \( y = x^3 \), just apply the rule directly.
• For complex functions such as \( y = (x^2 + 1)^{-1} \), first rewrite them in a power format and then proceed.

In our specific problem:

• The power rule helps in transforming \( y = \frac{1}{x^2 + 1} \) to a more workable form, \( y = (x^2 + 1)^{-1} \).
• From here, using the power rule gives the derivative, which is then used in the larger context of calculating the arc length.

Mastering the power rule simplifies the process of differentiation and opens a wider range of functions for calculus operation.