Question

Use the integration capabilities of a calculator to approximate the length of the curve.[T] \(r=\sin (3 \cos \theta)\) on the interval \(0 \leq \theta \leq \pi\)

Short answer

Use a calculator to approximate the integral for the curve from \( \theta = 0 \) to \( \theta = \pi \).

Step-by-step solution

Understand the Problem

We need to calculate the length of the curve described by the polar equation \( r = \sin(3 \cos(\theta)) \) from \( \theta = 0 \) to \( \theta = \pi \). To do this, we will use the integration capabilities of a calculator.

Use the Polar Arc Length Formula

The formula for the length of a curve in polar coordinates is \( L = \int_{a}^{b} \sqrt{ \left(\frac{dr}{d\theta}\right)^2 + r^2 } \, d\theta \), where \( r = \sin(3 \cos(\theta)) \) and the interval is \( 0 \leq \theta \leq \pi \).

Calculate \(\frac{dr}{d\theta}\)

First, find the derivative of \( r \) with respect to \( \theta \). If \( r = \sin(3 \cos(\theta)) \), then \( \frac{dr}{d\theta} = 3 \cos(3 \cos(\theta))(-\sin(\theta)) \).

Simplify the Arc Length Formula

Substitute \( \frac{dr}{d\theta} \) and \( r \) into the arc length formula: \[ L = \int_{0}^{\pi} \sqrt{(3 \cos(3 \cos(\theta))(-\sin(\theta)))^2 + (\sin(3 \cos(\theta)))^2 } \, d\theta \]

Use a Calculator for Numerical Integration

Input the equation from Step 4 into a calculator that can perform numerical integration. The calculator will approximate the integral value for the arc length of the curve.

Key concepts

Polar Coordinates

Polar coordinates offer a unique way to represent points on a plane, characterized by their distance from a fixed central point (usually called the pole) and the angle relative to a reference direction. Unlike Cartesian coordinates where each location is described using an x and y pair, polar coordinates use \(r, \theta\) — where \(r\) is the radial distance from the pole, and \(\theta\) is the angular coordinate represented by a direction from the initial side.

This system is especially useful when dealing with circular and angular motion, making calculations like curve length easier under certain circumstances. Curves described by polar equations, such as \(r = \sin(3 \cos(\theta))\), reveal lovely patterns and symmetries. In the context of the given problem, \(r\) depends on \(\theta\) through trigonometric functions, and the complexity of the equation shows a spiral-like path around the origin.

Understanding the fundamentals of polar coordinates is crucial when performing tasks involving motion or functions that are inherently circular or spiral, given their effectiveness in simplifying complex geometric and physical problems.

Numerical Integration

Numerical integration is a powerful tool for approximating the value of an integral when an analytical solution is difficult or impossible to find. It's essentially about estimating the area under a curve using a finite sum of values that approach the integral's true value.

Various methods are used for numerical integration, including the Trapezoidal Rule, Simpson's Rule, and the method of Riemann sums. Calculators and computer software can perform these calculations, making the integration process easier especially for complicated or long functions such as square roots of sum of squares as seen in the arc length problem above.

In this specific problem, to compute the curve's arc length for \(r = \sin(3 \cos(\theta))\), we deploy numerical integration to solve \(L = \int_{0}^{\pi} \sqrt{(\frac{dr}{d\theta})^2 + r^2} \, d\theta\) as the integral poses significant challenges for manual computation due to its form and limits. By leveraging technology, the cognitive workload associated with manual computation is greatly reduced.

Polar Arc Length Formula

The polar arc length formula is used when determining the length of a curve defined in polar coordinates between two angles. In this system, the length \(L\) of a polar curve can be derived from:

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

This formula considers both the change in the radial distance and the curvature of the path, calculating the true arc length across intervals of \(\theta\).

For our function \(r = \sin(3 \cos(\theta))\), this involves computing the derivative \(\frac{dr}{d\theta}\) first, and then substituting it back into the formula. Simplifying this formula and finding the result may require assistance from a calculator for accuracy and efficiency.

The polar arc length formula is essential when working with polar forms as it provides a direct method of calculating the path's length, capturing all nuances of any trigonometric distortions or transformations that may occur within the interval.