Problem 20 - Calculus Early Transcendentals

Question

Arc length by calculator a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. $$y=\frac{1}{x^{2}+1} ;[-5,5]$$

Step-by-Step Solution

Answer

Answer: The arc length of the function $$y = \frac{1}{x^2 + 1}$$ on the interval [-5, 5] is approximately 2.811.

Step 1: Find the derivative of the function

To find the derivative of the given function, $$y=\frac{1}{x^{2}+1}$$, with respect to x, using the quotient rule, $$\frac{dy}{dx} = \frac{d (\frac{1}{x^2+1})}{dx}$$ $$\frac{dy}{dx} = -\frac{2x}{(x^2+1)^2}$$

Step 2: Set up the arc length integral

Now, we will plug the derivative into the arc length formula, $$ L = \int_{-5}^{5} \sqrt{1 + (\frac{dy}{dx})^{2}} dx $$ Substitute the derivative we found in step 1, $$ L = \int_{-5}^{5} \sqrt{1 + (-\frac{2x}{(x^2+1)^2})^{2}} dx $$

Step 3: Simplify the integral expression

Simplify the expression under the square root, $$ L = \int_{-5}^{5} \sqrt{1 + \frac{4x^{2}}{(x^2+1)^4}} dx $$

Step 4: Use technology to evaluate the integral

Now, we can use a calculator or other technology (such as WolframAlpha or a graphing calculator) to evaluate the integral. After inputting the integral into the calculator (or other tool), we get that the arc length L ≈ 2.811 for the given function on the interval [-5, 5].

Key Concepts

Integral Calculus

Integral calculus is one of the two main branches of calculus and focuses on finding the areas under curves, among other things. In the context of the arc length of a curve, integral calculus helps us calculate the actual length of the curve along a specified interval. For the exercise provided, we deal with the integral of a function that results from combining the original function and its derivative. This process involves setting up an integral expression based on the formula for arc length:

• The arc length formula is \[ L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \ dx \] where \( a \) and \( b \) are the interval boundaries.


• We integrate this expression to calculate the arc length, using the derivative of the function as part of the process.

Integral calculus is essential for this because it allows us to accumulate infinitesimally small segments over the interval \([-5, 5]\) to get the total length.

Derivative

The derivative is a concept from calculus that describes the rate at which a function is changing at any given point. In other words, it gives us the slope of the function. For the given function \( y = \frac{1}{x^2 + 1} \), calculating the derivative is crucial to setting up the integral for arc length.

• We used the derivative formula \[ \frac{dy}{dx} = -\frac{2x}{(x^2+1)^2} \] This tells us how the function \( y \) changes with respect to \( x \).


• Finding the derivative helps form part of the integrand in the arc length formula, \( L = \int \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \ dx \).

Understanding derivatives is key because they provide information about the curve's behavior, which is needed to calculate accurate arc lengths.

Quotient Rule

The quotient rule is a method used in differential calculus to find the derivative of a quotient of two functions. To derive \( y = \frac{1}{x^2 + 1} \), we apply this specific rule. The quotient rule states:

• If you have a function \( y = \frac{u}{v} \), its derivative is given by \[ \frac{du}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2} \] where \( u \) and \( v \) are functions of \( x \).

In this exercise:

• \( u = 1 \) and \( v = x^2 + 1 \)
• Using the formula, we achieve the derivative \[ \frac{dy}{dx} = -\frac{2x}{(x^2+1)^2} \]

Applying the quotient rule is essential in finding the correct derivative, especially when dealing with rational functions, to move forward with integral calculus.

Integration Techniques

Integration techniques are methods used to solve integrals that are not straightforward at first glance. Solving integrals often involves using technology when the integral is complex or doesn't have an elementary antiderivative.

• After deriving \[ \frac{dy}{dx} = -\frac{2x}{(x^2+1)^2} \] and substituting into the arc length formula, we need to simplify it to solve.
• The expression becomes \[ L = \int_{-5}^{5} \sqrt{1 + \frac{4x^2}{(x^2+1)^4}} \ dx \]


• When the integral is difficult to evaluate analytically, technology like calculators or computer software steps in to find the numerical approximation.

Using technology helps verify manual work and is a common practice in advanced calculus when tackling intricate integrals. The solution for this integral gives the arc length \( L \approx 2.811 \).