Problem 18 - Thomas Calculus

Question

In Exercises \(13-20,\) do the following. \begin{equation}\begin{array}{l}{\text { a. Set up an integral for the length of the curve. }} \\ {\text { b. Graph the curve to see what it looks like. }} \\\ {\text { c. Use your grapher's or computer's integral evaluator to find }} \\\ {\text { the curve's length numerically. }}\end{array}\end{equation} $$y=\sin x-x \cos x, \quad 0 \leq x \leq \pi$$

Step-by-Step Solution

Answer

Set up the integral, graph the curve, and numerically evaluate \(L \approx 4.82\).

Step 1: Identify the formula for arc length

The formula for the arc length of a curve \(y = f(x)\) from \(x = a\) to \(x = b\) is given by: \[ L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \] where \(\frac{dy}{dx}\) is the derivative of \(y\) with respect to \(x\).

Step 2: Find the derivative of the function

Given the function \( y = \sin x - x \cos x \), find its derivative: \( \frac{dy}{dx} = \cos x + x \sin x - \cos x = x \sin x \).

Step 3: Plug the derivative into the arc length formula

Substitute \( \frac{dy}{dx} = x \sin x \) into the arc length formula: \[ L = \int_0^{\pi} \sqrt{1 + (x \sin x)^2} \, dx \].

Step 4: Graph the given curve

Plot the function \( y = \sin x - x \cos x \) over the interval \( 0 \leq x \leq \pi \) to see its shape. The curve will illustrate wave-like behavior within the given bounds.

Step 5: Evaluate the integral numerically

Use a numerical method or calculator to evaluate the integral \( L = \int_0^{\pi} \sqrt{1 + (x \sin x)^2} \, dx \) to find the arc length. This step typically involves using a graphing calculator or software capable of numerical integration.

Key Concepts

Integral Calculation

Integral calculation is a fundamental concept in calculus that allows us to find the total amount, such as area or arc length, over a specified interval. In our scenario, we are interested in calculating the arc length of the curve described by the function \( y = \sin x - x \cos x \) over the interval \( 0 \leq x \leq \pi \).
To do this, we use the arc length formula:

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

This formula helps compute the length of the curve from \( x = a \) to \( x = b \).
By substituting the derivative of the function into our formula, we get the exact integral expression that needs to be evaluated. Integrals, by nature, provide the sum of infinitesimally small quantities, which, when handled properly, give us the length of a curve smoothly. Remember, evaluating integrals often involves finding an antiderivative or using numerical methods if the integral is too complex for basic solutions.

Derivative of a Function

Understanding derivatives is crucial in expressing the rate of change of a function. Here, derivatives play an essential role in setting up the integral for the arc length calculation.
Given our function \( y = \sin x - x \cos x \), the first task is to determine its derivative. This gives us:

• \( \frac{dy}{dx} = \cos x + x \sin x - \cos x \)
• Simplifying it further, we arrive at \( \frac{dy}{dx} = x \sin x \)

The derivative, \( \frac{dy}{dx} \), measures the slope of the tangent line at any point in the curve, providing us with necessary information about how the function changes.
Plugging this derivative back into our arc length formula is what allows us to calculate the overall arc length. In calculus, derivatives not only tell us about the rate of change but also help us reconstruct whole lengths and areas when integrated adequately.

Numerical Integration

Numerical integration comes into play when integrals are too complex to solve via traditional analytical methods. Techniques such as Simpson's Rule, Trapezoidal Rule, or using sophisticated calculators can estimate the value of integrals.
Given our integral \( L = \int_0^{\pi} \sqrt{1 + (x \sin x)^2} \, dx \), obtaining an exact solution analytically might prove cumbersome.
This is where numerical methods shine. By breaking down the integral into smaller parts, these techniques approximate the area under the curve with a high degree of accuracy.

• These methods require partitioning the interval into small sub-intervals.
• Calculating endpoints or midpoint function values, depending on the method.
• Summing these values up to find the total integral value.

Utilizing graphing calculators or software simplifies this exercise and provides exact or near-exact results that are suitable for practical purposes.

Curve Graphing

Graphing curves visually brings to life the behavior of mathematical functions.
Our function, \( y = \sin x - x \cos x \), illustrates how concepts such as trigonometry combine to produce wave-like graphs. Graphing these curves can be accomplished with graphing calculators or computer software, especially over the interval \( 0 \leq x \leq \pi \).
Visualization helps in:

• Understanding the nature of points where the curve changes direction.
• Identifying local maxima and minima of the curve.
• Checking whether the curve's plotted shape aligns with expected mathematical behavior.

Graphing is critical for curve analysis as it verifies calculations and assumptions made in both integration and differentiation processes, offering a helpful visual checkpoint for mathematical thought and problem-solving.