Problem 22 - Thomas Calculus

Question

In Exercises \(17-24,\) do the following. 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 length of the curve is approximately calculated using numerical integration.

Step 1: Calculate Derivative

First, to find the length of the curve given by \(y = \sin x - x \cos x\) from \(x = 0\) to \(x = \pi\), we need the derivative \(y'\). Differentiate with respect to \(x\): \(y = \sin x - x \cos x\). Thus, \(y' = \cos x - (\cos x - x \sin x) = x \sin x\).

Step 2: Set Up the Integral

The formula for the arc length of a curve \(y = f(x)\) from \(x = a\) to \(x = b\) is \(\int_a^b \sqrt{1 + (f'(x))^2} \, dx\). Here, \(f'(x) = x \sin x\). Thus, the integral becomes \(\int_0^{\pi} \sqrt{1 + (x \sin x)^2} \, dx\).

Step 3: Graph the Curve

To visualize \(y = \sin x - x \cos x\) for \(0 \leq x \leq \pi\), graph the function. This can be done using graphing software or a graphing calculator. The curve resembles a sine wave but is adjusted by the term \(-x \cos x\).

Step 4: Numerical Evaluation

Use a graphing calculator or computer software that supports numerical integration to approximate the value of the integral \(\int_0^{\pi} \sqrt{1 + (x \sin x)^2} \, dx\). This gives the numerical value of the curve's length.

Key Concepts

Arc Length

When it comes to finding the arc length of a curve, calculus provides a powerful tool that involves integration. Arc length essentially measures the distance along a path, which can be straight or curved. To compute the arc length of a function like \(y = \sin x - x \cos x\) over the interval \(0 \leq x \leq \pi\), you would set up an integral.To determine this, we use the formula:\[L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx\]This means you'll need the first derivative \(f'(x)\), because the arc length formula involves integrating the square root of \(1 + (f'(x))^2\). For our function, \(f'(x) = x \sin x\), which is obtained by differentiating the given equation. Substituting this into the integral gives you the setup needed to find the arc length. Remember, this process will give you the arc length by integrating over the specified interval.

Differentiation

Differentiation is one of the cornerstones of calculus and is used to find the rate at which a function is changing at any given point. In the context of finding arc lengths, differentiation is crucial because it provides the derivative needed in the arc length formula!The derivative of a function, denoted as \(y'\) or \(f'(x)\), gives you the slope of the tangent line to the curve at any point \(x\). For \(y = \sin x - x \cos x\), applying the rules of differentiation involves:

• Using the product rule for the term \(-x \cos x\), which results in a more complex derivative involving both \(\cos x\) and \(\sin x\).
• Simplifying the derivative to find \(f'(x) = x \sin x\).

Understanding differentiation fully will allow you to tackle not only arc length problems but also a wide range of applications where rates of change are involved.

Numerical Integration

Sometimes, solving integrals analytically, especially when dealing with complex functions, can be difficult or impossible to do by hand. This is where numerical integration methods come into play. They provide approximations of the integral's value, which often suffices for practical purposes.In the case of \(\int_0^{\pi} \sqrt{1 + (x \sin x)^2} \, dx\), using numerical integration will help us estimate this integral over the interval \(0\) to \(\pi\). Tools such as graphing calculators or computer software handle this by employing techniques like the trapezoidal rule or Simpson's rule to approximate the area under the curve defined by our integrand.An accurate numerical integration technique will allow you to effectively find approximate solutions when exact evaluation isn't feasible.