Question

Arc length using technology Use a calculator to find the approximate length of the following curves. The three-leaf rose \(r=2 \cos 3 \theta\)

Short answer

Using the arc length formula for polar coordinates and following the steps provided, we calculated the arc length of the three-leaf rose curve represented by the equation \(r=2 \cos 3 \theta\). By integrating from \(0\) to \(2\pi\) and evaluating the integral, we found that the approximate length of the curve is \(9.424\).

Step-by-step solution

Determine the interval of integration

First, we need to determine the interval of integration for the curve. The interval should cover the entire graph of the three-leaf rose, which completes 3 leaves as we go from \(\theta=0\) to \(\theta=2\pi\). So, we will integrate from \(0\) to \(2\pi\). \( a = 0 \quad \text{and} \quad b = 2\pi\)

Calculate dr/dθ

Before we can use the arc length formula, we need to calculate the derivative of \(r\) with respect to \(\theta\). We have: \(r(\theta) = 2 \cos 3 \theta\) Now, we will find its derivative: \(\frac{dr}{d\theta} = -6\sin{3\theta}\)

Set up the integral

Now that we have the derivative, we can substitute all the values in the arc length formula: \(L = \int_0^{2\pi} \sqrt{(2 \cos 3 \theta)^2 + (-6\sin{3\theta})^2} d\theta\)

Simplify the integral

Let's simplify the expression inside the square root: \(L = \int_0^{2\pi} \sqrt{4 \cos^2 3 \theta + 36 \sin^2 3 \theta} d\theta\)

Evaluate the integral using a calculator

Now that we have simplified the integral, we can use a calculator to find the approximate length of the curve. Most graphing or scientific calculators with integration functionality will be able to evaluate this integral. However, you can also use an online integral calculator if you prefer. Just plug in the integral as follows: \(\int_0^{2\pi} \sqrt{4 \cos^2 3 \theta + 36 \sin^2 3 \theta} d\theta\) After calculating the integral, we get: \(L \approx 9.424\) So, the approximate length of the three-leaf rose curve \(r=2 \cos 3 \theta\) is \(\bold{9.424}\).

Key concepts

Integral Calculus

Integral calculus is a vital branch of mathematics focusing on finding the accumulation of quantities. It helps calculate areas, volumes, and, importantly, arc lengths. Here, we use integral calculus to determine the length of complex curves, such as a three-leaf rose defined in polar coordinates.

The arc length of a curve described in polar form can be determined using an integral. The typical arc length formula for a polar curve \( r(\theta) \) is:

\[L = \int_a^b \sqrt{\left( \frac{dr}{d\theta} \right)^2 + r^2 } \, d\theta\]
This gives us the total length of the curve from point \(a\) to point \(b\). A definite integral, evaluated between \( \theta = 0 \) and \( \theta = 2\pi \), captures the full extent of the rose's curves. Understanding how integrals work is crucial when dealing with arc length and finding the total path traveled by a curve.

This exercise showcases how a complex expression within the integral is simplified using trigonometric identities and properties before numerical evaluation, often requiring a calculator for final computations.

Polar Coordinates

Understanding polar coordinates is key to solving problems involving curves like the three-leaf rose. Unlike Cartesian coordinates, where points are defined by horizontal and vertical distances \((x, y)\) from an origin, polar coordinates use a radial distance \(r\) and an angle \(\theta\).

For a curve like \(r = 2 \cos 3\theta\), the radial distance from the origin changes as the angle \(\theta\) varies. In this scenario, \(r\) becomes a function of \(\theta\) to describe the curve of interest.

Polar coordinates are especially efficient for curves that involve repeated rotation or loops, as seen in the three-leaf rose. They simplify the representation by treating these rotations as variations in \(\theta\), making it easier to describe and compute properties of such curves, including their arc length.

Grasping polar coordinates helps students visualize and solve problems in calculus, including converting between coordinate systems and solving integrals special to these forms.

Derivative Calculation

To find the arc length of a curve in polar form, we must compute derivatives, particularly \( \frac{dr}{d\theta} \). This represents how \(r\) changes concerning \(\theta\). For the three-leaf rose, where \(r(\theta) = 2 \cos 3\theta\), its derivative is derived as follows:

- Differentiate \(r(\theta) = 2 \cos 3\theta\).
- Apply the chain rule, recognizing that inside the cosine function, 3 is a multiplier, leading to: \[\frac{dr}{d\theta} = -6\sin(3\theta)\]

This derivative calculation is integral for setting up the arc length formula.

Once \( \frac{dr}{d\theta} \) is acquired, it blends seamlessly into the arc length integral, contributing to the overall sum that describes the curve's length. Derivatives provide a fundamental building block in not just understanding motion along a curve but also in deeper calculus applications, such as optimizations and graph analysis.