Question

Use the integration capabilities of a calculator to approximate the length of the curve.[T] \(r=2 \theta^{2}\) on the interval \(0 \leq \theta \leq \pi\)

Short answer

Approximate the integral \( \int_{0}^{\pi} 2\theta\sqrt{4 + \theta^2} \, d\theta \) using a calculator to find the curve length.

Step-by-step solution

Understand the Curve Equation

We need to find the length of the polar curve given by \( r = 2 \theta^2 \) from \( \theta = 0 \) to \( \theta = \pi \). First, recognize that this is a polar equation where \( r \) is expressed as a function of \( \theta \).

Use the Curve Length Formula for Polar Coordinates

The length of a curve in polar coordinates is given by the integral\[ L = \int_{a}^{b} \sqrt{ \left(\frac{dr}{d\theta}\right)^2 + r^2 } \, d\theta \]where in this exercise, \( a = 0 \) and \( b = \pi \).

Find the Derivative of \( r \) with Respect to \( \theta \)

The given polar equation is \( r = 2 \theta^2 \). Find \( \frac{dr}{d\theta} \) by differentiating: \( \frac{dr}{d\theta} = \frac{d}{d\theta}(2\theta^2) = 4\theta \).

Substitute into the Length Formula

Substitute \( r = 2\theta^2 \) and \( \frac{dr}{d\theta} = 4\theta \) into the length formula:\[ L = \int_{0}^{\pi} \sqrt{ (4\theta)^2 + (2\theta^2)^2 } \, d\theta \]. Simplifying the integrand gives \[ L = \int_{0}^{\pi} \sqrt{ 16\theta^2 + 4\theta^4 } \, d\theta \].

Simplify and Evaluate the Integral

Further simplify the integrand: \(L = \int_{0}^{\pi} \sqrt{ 4\theta^2 (4 + \theta^2) } \, d\theta \), which simplifies to \(L = \int_{0}^{\pi} 2\theta\sqrt{4 + \theta^2} \, d\theta \). Use a calculator to approximate this integral.

Use a Calculator for the Integration

Use a calculator capable of numerical integration to find the approximate value of \( \int_{0}^{\pi} 2\theta\sqrt{4 + \theta^2} \, d\theta \). Input the function directly into the calculator, ensuring the correct limits and function are used.

Key concepts

Curve Length

In the context of polar coordinates, determining the length of a curve involves understanding how the path stretches over a given interval. Polar coordinates describe curves using a radial distance, denoted as \( r \), that varies with the angle \( \theta \). This description contrasts with Cartesian coordinates, where curves are typically defined as \( y \) in terms of \( x \).
You can calculate the length of a curve described in polar coordinates using the formula for curve length, given by:

• \[ L = \int_{a}^{b} \sqrt{ \left(\frac{dr}{d\theta}\right)^2 + r^2 } \, d\theta \]

Here, \( a \) and \( b \) are the limits of \( \theta \) over which you are measuring the curve. The formula takes into account both the change of radius and the angle over the interval, integrating these changes to yield the total length of the curve. This formula is key in finding how long the curve extends as the polar angle changes.

Integration

Integration is a fundamental tool in calculus used to sum infinitesimal changes. In the context of this exercise, we use integration to calculate the length of a curve as it extends over an interval of \( \theta \).
The integration process involves taking a continuous curve and breaking it down into infinitely small segments to sum up their lengths. In particular, when utilizing the curve length formula, understanding the integral means seeing how small changes in \( \theta \) contribute to the overall curve length.
This integral

• \[ L = \int_{0}^{\pi} 2\theta\sqrt{4 + \theta^2} \, d\theta \]

represents the summation of the tiny distances as \( \theta \) progresses from \( 0 \) to \( \pi \). The specific function \( 2\theta\sqrt{4 + \theta^2} \) inside the integral accounts for how the curve's shape changes along this range. Understanding integration in this way helps in interpreting the total length of curves in polar coordinates.

Numerical Approximation

Sometimes, exact integration is complex or impossible to compute analytically, which is where numerical approximation comes in handy. In this exercise, a calculator is used to approximate the integral solution, which in turn approximates the curve's length.
Numerical approximation techniques, such as Simpson's Rule or the Trapezoidal Rule, essentially break the integral into manageable sections that are easier to evaluate. These methods approximate the area under the curve or, in our case, the total curve length, by summing up these smaller segments.
When using a calculator, it's vital to:

• Ensure the function being integrated is entered correctly.
• Set the correct limits of integration, \( 0 \) to \( \pi \) in this example.
• Understand the calculator’s output as an approximation, not an exact value.

This approach is quite useful, especially when dealing with complex functions like \( 2\theta\sqrt{4 + \theta^2} \), where algebraic solutions may not be feasible. Numerical methods allow for practical solutions with a desired degree of accuracy.