Problem 18 - Thomas Calculus in SI Units

Question

a. Set up an integral for the length of the curve. b. Graph the curve to see what it looks like. c. Use your grapher's or computer's integral evaluator to find the curve's length numerically. $$y=\sin x-x \cos x, \quad 0 \leq x \leq \pi$$

Step-by-Step Solution

Answer

The curve's length is approximately the value found numerically from the integral evaluation.

Step 1: Derivative of the Function

To find the length of the curve, we first need to determine the derivative of \( y = \sin x - x \cos x \). Applying the derivative rules, the derivative is \( y' = \cos x - \cos x + x \sin x = x \sin x \).

Step 2: Applying the Arc Length Formula

The formula for the arc length \( L \) of a curve given by \( y = f(x) \) from \( x = a \) to \( x = b \) is \( L = \int_a^b \sqrt{1 + \left( f'(x) \right)^2} \, dx \). Substituting the derivative found in Step 1, we set up the integral: \( L = \int_0^\pi \sqrt{1 + (x \sin x)^2} \, dx \).

Step 3: Visualizing the Curve

To graph \( y = \sin x - x \cos x \) over the interval \( 0 \leq x \leq \pi \), plot the function using graphing software or a calculator. Observing this graph helps understand the behavior of the function and confirms our integral setup corresponds to this visual representation.

Step 4: Numerical Evaluation of the Integral

Using computational tools such as a graphing calculator or software (e.g., Desmos, Wolfram Alpha), evaluate the integral \( \int_0^\pi \sqrt{1 + (x \sin x)^2} \, dx \) to find an approximate numerical value for the curve's length.

Key Concepts

Curve Graphing

When we talk about graphing a curve, we are essentially trying to visualize how a function behaves over a specific interval. In this exercise, the function we're dealing with is \( y = \sin x - x \cos x \) from \( 0 \) to \( \pi \). Graphing the function helps us see its shape, which can provide insights into how the curve behaves.
To graph this curve, you can use graphing calculators or software like Desmos. Such tools offer an easy way to input the function and instantly see a visual representation.
When you plot \( y = \sin x - x \cos x \) over the specified interval, you'll notice how the sine and cosine components influence the curve's overall shape. This graphing step not only ensures you're solving the right problem, but it provides a firm foundation upon which to understand further calculations, like setting up the right integral for curve length.

Integral Setup

Setting up the correct integral is a key step in finding the length of a curve. In this context, the function \( y = \sin x - x \cos x \) requires us to use integral calculus to determine its arc length.
The arc length formula is a powerful tool that allows us to calculate the precise length of any smooth curve. This formula is given by:

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

Here, \( \frac{dy}{dx} \) is the derivative of the function, which tells us about the slope of the curve at any point.
For our function, the derivative comes out to be \( \frac{dy}{dx} = x \sin x \). Substituting this into the arc length formula, we set up the integral as:

• \[ L = \int_0^\pi \sqrt{1 + (x \sin x)^2} \, dx \]

This integral will give us the total length of the curve over the range from \( 0 \) to \( \pi \). Setting up the correct integral is crucial as an incorrect setup could lead to major errors in further calculations.

Numerical Integration

Consider numerical integration as a tool for approximating the value of complex integrals that are challenging to solve analytically. When you have an integral like \( \int_0^\pi \sqrt{1 + (x \sin x)^2} \, dx \), manual calculations can be quite intricate.
Instead, numerical techniques approximate this value with acceptable precision. Tools like a graphing calculator or computational software (e.g., Wolfram Alpha) can automate this process.
These tools typically use methods like the Trapezoidal Rule or Simpson's Rule to break down the area under the curve into simpler shapes (trapezoids or parabolas). They calculate each piece's area and sum these to approximate the full integral value.
Using these computational aids not only saves time but also reduces manual errors, revealing the curve's length with a high degree of accuracy. It's an essential practice when dealing with integrals that do not have simple analytical solutions.