Problem 18 - Thomas Calculus

Question

In Exercises \(13-20,\) 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

Integral evaluates to the curve's approximate length from \( x = 0 \) to \( \pi \).

Step 1: Set Up the Integral for the Curve Length

To calculate the length of the curve \( y = \sin x - x \cos x \) from \( x = 0 \) to \( x = \pi \), we use the arc length formula: \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]First, find the derivative \( \frac{dy}{dx} \):\[ \frac{dy}{dx} = \cos x + x \sin x \]Now set up the integral:\[ L = \int_{0}^{\pi} \sqrt{1 + (\cos x + x \sin x)^2} \, dx \]

Step 2: Graph the Curve

To visualize the curve \( y = \sin x - x \cos x \), plot it for the domain \( 0 \leq x \leq \pi \) using graphing software or a graphing calculator. Observe the wave-like nature of the sine function modified by the linear term \( -x \cos x \). This helps in ensuring the limits and the function are set correctly before evaluating any integrals.

Step 3: Evaluate the Integral Numerically

Use a graphing calculator or computer integral evaluation tool to calculate the length of the curve from \( x = 0 \) to \( x = \pi \) using the integral formula derived:\[ L = \int_{0}^{\pi} \sqrt{1 + (\cos x + x \sin x)^2} \, dx \]Numerical evaluation will give the approximate length of the curve, which can be evaluated using software like Mathematica, MATLAB, or online integral calculators.

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics that deals with the accumulation of quantities, often represented through the process of integration. In this context, we are using it to find the arc length of a curve described by the function \(y = \sin x - x \cos x\) over the interval from \(x = 0\) to \(x = \pi\).
To find the length of a curve, the formula used is \( L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \). This formula comes from considering small segments of the curve as lines and summing their lengths.
First, we find the derivative \( \frac{dy}{dx} \) of the function, which in this case is \( \cos x + x \sin x \). Substitute this into the arc length formula to get \( L = \int_{0}^{\pi} \sqrt{1 + (\cos x + x \sin x)^2} \, dx \).

• This integral was set up to find the exact length of the curve.
• Integral calculus is essential here because it lets us calculate lengths that are not just straight lines.

Remember, the process involves finding the derivative and then substituting it into the arc length formula for integration.

Curve Graphing

Curve graphing is important in visualizing the characteristics and behavior of mathematical functions. Here, the function \(y = \sin x - x \cos x\) is graphed over the interval \(0 \leq x \leq \pi\). This step is vital for understanding how the function behaves and ensuring the calculations are correct.
By graphing:\

• You can visually confirm the shape of the curve.
• Identify the impact of each term, like the wave-like behavior due to \(\sin x\) and the modification from \(-x \cos x\).
• Ensure that limits and the function are correct before numerical calculation.

Graphing with a graphing calculator or software assists in this process, offering a clear picture of the curve's profile. It allows a better grasp of why certain calculations follow, especially in complex functions.

Numerical Integration

Not all integrals have a straightforward analytical solution, which is where numerical integration comes into play. In our example, calculating \( \int_{0}^{\pi} \sqrt{1 + (\cos x + x \sin x)^2} \, dx \) precisely is complex, and often, a numerical method is employed for approximation.
Numerical integration methods like the trapezoidal rule or Simpson's rule provide approximate solutions by summing the values of the function at certain points:

• Graphing calculators or software tools like MATLAB or Mathematica can perform numerical integrations efficiently.
• These tools break the integral into small segments, estimating the curve's length by adding up these small lengths.

The result is an approximate value of the curve's length, which is very useful when exact integration isn’t feasible analytically. This step is often the last part in confirming the total arc length of the curve.