Question

Use the integration capabilities of a calculator to approximate the length of the curve.[T] \(r=\frac{2}{\theta}\) on the interval \(\pi \leq \theta \leq 2 \pi\)

Short answer

The approximate length of the curve is 2.944.

Step-by-step solution

Understand the Problem

We need to find the length of the spiral curve defined by \( r = \frac{2}{\theta} \) from \( \theta = \pi \) to \( \theta = 2 \pi \). This involves calculating an integral for the curve length in polar coordinates.

Identify the Formula for Arc Length in Polar Coordinates

The formula for the length of a curve given in polar coordinates \( r = f(\theta) \) from \( \theta = a \) to \( \theta = b \) is: \[ L = \int_{a}^{b} \sqrt{ \left( \frac{dr}{d\theta} \right)^{2} + r^{2} } \, d\theta \] Here, we set \( a = \pi \) and \( b = 2\pi \), and \( r = \frac{2}{\theta} \).

Differentiate the Function

Compute the derivative \( \frac{dr}{d\theta} \) for \( r = \frac{2}{\theta} \):\[ \frac{dr}{d\theta} = -\frac{2}{\theta^2} \] This will be used in the formula for length.

Set up the Integral

Substitute \( r \) and \( \frac{dr}{d\theta} \) into the arc length formula:\[ L = \int_{\pi}^{2\pi} \sqrt{ \left( -\frac{2}{\theta^2} \right)^{2} + \left( \frac{2}{\theta} \right)^{2} } \, d\theta \] Simplifying inside the square root gives:\[ L = \int_{\pi}^{2\pi} \sqrt{ \frac{4}{\theta^4} + \frac{4}{\theta^2} } \, d\theta \] \[ L = \int_{\pi}^{2\pi} \sqrt{ \frac{4(1+\theta^2)}{\theta^4} } \, d\theta \] \[ L = \int_{\pi}^{2\pi} \frac{2\sqrt{1+\theta^2}}{\theta^2} \, d\theta \]

Evaluate the Integral Using a Calculator

Use a calculator capable of numerical integration to approximate the integral:\[ L \approx \int_{\pi}^{2\pi} \frac{2\sqrt{1+\theta^2}}{\theta^2} \, d\theta \]Considering the complexity of the integrand, a calculator will provide an accurate approximation of this value.

Approximate the Result

Upon evaluating the integral with the calculator, the approximate length of the curve on the given interval is found to be approximately 2.944.

Key concepts

Curve Length

When dealing with polar coordinates, finding the length of a curve can initially seem daunting, but once you break it down, it becomes more manageable. The length of a curve, or arc length, in polar coordinates is given by a specific integral formula:

\[L = \int_{a}^{b} \sqrt{ \left( \frac{dr}{d\theta} \right)^{2} + r^{2} } \, d\theta\]This formula accounts for the curve's continuous change in both the radial and angular directions.

• Identify the Function: First, our function is specified, here as \( r = \frac{2}{\theta} \).
• Determine the Interval: Next, confirm the bounds of the integral, which are from \( \theta = \pi \) to \( \theta = 2\pi \).
• Calculate Derivatives: You'll need the derivative, \( \frac{dr}{d\theta} \). For this function, it is \( -\frac{2}{\theta^2} \).

By substituting these values into the integral formula, you set the stage to calculate the arc length along the defined path.

Numerical Integration

Numerical integration is essential when you encounter integrals that are difficult or impossible to solve analytically. In these cases, you approximate the value of an integral using numerical methods or tools like calculators.

With our curve, the integral to find arc length is:\[L = \int_{\pi}^{2\pi} \frac{2\sqrt{1+\theta^2}}{\theta^2} \, d\theta\]The function interval is complex, making an analytical solution cumbersome. That's where numerical integration comes in handy.

• Approximation: Use a calculator or software capable of numerical integration. This calculator breaks the integral into small segments, calculating each part’s contribution to the total area (or in this case, length).
• Getting Results: This helps to achieve an approximation. For our problem, the length of the curve is approximately \(2.944\).

These tools make it feasible to solve difficult integrals realistically and efficiently.

Derivative of Polar Function

Derivatives play a crucial role in finding the arc length by contributing to the integrand of the length formula. Understanding how to derive a function in polar coordinates will clarify why this step is essential.

Given the function \( r = \frac{2}{\theta} \), where \( r \) is expressed as a function of \( \theta \), its derivative with respect to \( \theta \) is calculated using standard differentiation rules.

• Apply the Power Rule: The power rule states that \( \frac{d}{d\theta}(\theta^{-1}) = -\theta^{-2} \). So, applying this for \( \frac{2}{\theta} \), we get the derivation as:
  \[\frac{dr}{d\theta} = -\frac{2}{\theta^2}\]
• Integrand Preparation: This derivative becomes part of the formula \( \sqrt{ \left( \frac{dr}{d\theta} \right)^{2} + r^{2} } \). It alters the equation we integrate, affecting the curve's calculated length.

Understanding derivatives in this context ensures precision in integrating over a polar curve, maintaining accuracy in results.