Question

Arc length using technology Use a calculator to find the approximate length of the following curves. $$\text { The lemniscate } r^{2}=6 \sin 2 \theta$$

Short answer

Answer: The approximate length of the curve is 9.688.

Step-by-step solution

Calculate the derivatives of $$r^2$$ with respect to $$\theta$$.

We are given the polar equation $$r^2 = 6 \sin 2 \theta$$, and we need to find $$\frac{dr}{d\theta}$$. First, we differentiate both sides of the equation with respect to $$\theta$$: $$2r\frac{dr}{d\theta} = 12\cos 2\theta$$ Now, we need to isolate $$\frac{dr}{d\theta}$$: $$\frac{dr}{d\theta} = \frac{6\cos 2\theta}{r}$$

Plug our derivatives and our polar equation into the arc length formula.

We will substitute the expression for $$r^2$$ and $$\frac{dr}{d\theta}$$ into the arc length formula: $$L = \int_{\alpha}^{\beta} \sqrt{6 \sin 2 \theta + \Big(\frac{6\cos 2\theta}{\sqrt{6 \sin 2 \theta}}\Big)^2} d\theta$$

Determine the appropriate integration bounds.

A lemniscate has symmetry in both the x and y axes. We can take advantage of this symmetry and just find the arc length of one loop and then multiply it by 2. To do this, we will find the arc length between points where $$\theta = 0$$ and $$\theta = \frac{\pi}{2}$$, since the lemniscate intersects the x-axis at these points: $$L = 2\int_{0}^{\frac{\pi}{2}} \sqrt{6 \sin 2 \theta + \Big(\frac{6\cos 2\theta}{\sqrt{6 \sin 2 \theta}}\Big)^2} d\theta$$

Use a calculator to evaluate the integral and get the approximate arc length.

Now, using a calculator to evaluate the integral, we find the approximate arc length of the lemniscate: $$L \approx 2\int_{0}^{\frac{\pi}{2}} \sqrt{6 \sin(2 \theta) + \Big(\frac{6\cos(2 \theta)}{\sqrt{6 \sin(2 \theta)}}\Big)^2} d\theta \approx 9.688$$ So, the approximate length of the curve is 9.688.

Key concepts

Lemniscate

The lemniscate is a fascinating mathematical curve that resembles the shape of an infinity symbol (∞) or a figure eight. It is known for its intriguing properties in geometry and calculus. The particular lemniscate described by the equation \(r^2 = 6 \sin 2\theta\) in polar coordinates has some unique characteristics. By its nature, a lemniscate is symmetric about the origin, meaning it looks the same on both sides of every axis crossing through the origin.
The structure of a lemniscate in polar coordinates relies heavily on polar equations. These curves have two loops, and depending on the parameters of the equation, the size and shape of these loops can vary. Such properties often make the lemniscate a point of interest in advanced mathematics studies. To find specifics about any lemniscate, often calculus is applied, as seen in the determination of arc lengths or enclosed areas.

Polar Coordinates

Polar coordinates provide a different method for describing the location of a point in a plane, using a distance and an angle from a fixed point, usually the origin. They are particularly helpful in dealing with complex curves like the lemniscate, where the form in rectangular (Cartesian) coordinates would be awkward.
In the polar coordinate system, each point on a plane is determined by a radius \(r\) and an angle \(\theta\). This system is invaluable when working with circular or rotational symmetries, commonly found in nature and various applications in physics and engineering. Understanding polar coordinates allows us to visualize and analyze these curves in a more simplified manner by utilizing the angle and the distance from the center.

Derivative

When dealing with polar equations, such as \(r^2 = 6 \sin 2\theta\), the derivative \(\frac{dr}{d\theta}\) plays a crucial role in determining characteristics like arc length. Derivatives help to find how a quantity changes as another quantity changes, providing insights into the curvature and dynamics of a curve.
In the given example, the derivative \(\frac{dr}{d\theta} = \frac{6\cos 2\theta}{r}\) was obtained through implicit differentiation, which is a method used to differentiate equations where one variable is a function of another. This process is essential for further calculations and understanding how the lemniscate behaves as \(\theta\) changes.

Integral Calculation

The calculation of an integral is a fundamental technique in calculus used to find areas, volumes, central points, and many other important measurements. When trying to find the arc length of the lemniscate \(r^2 = 6 \sin 2\theta\), we use an integral formula to sum up infinitely small pieces of the curve.
The arc length formula in polar coordinates: \[L = \int_{\alpha}^{\beta} \sqrt{\left(\frac{dr}{d\theta}\right)^2 + r^2} d\theta\] allows for the evaluation of the curve's total length between limits \(\alpha\) and \(\beta\). Using symmetry properties, calculations can be simplified. For this problem, the symmetry of the lemniscate allowed us to compute the integral over half of the curve and then double the result to find the total arc length. This simplifies the computation process significantly, yielding an approximate arc length of 9.688 upon evaluation with a calculator.