Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the length of the curve with the given parametric equations (a) \(x=\sin \theta, y=\cos \theta\) for \(0 \leq \theta \leq 2 \pi\) (b) \(x=\sin 3 \theta, y=\cos 3 \theta\) for \(0 \leq \theta \leq 2 \pi\) (c) Explain why the lengths in parts (a) and (b) are not equal.

Step-by-Step Solution

Answer

Part (a) length is \(2\pi\) and part (b) length is \(6\pi\). Part (b) is traced three times around the unit circle.

Step 1: Recall the Formula for Length of a Parametric Curve

The length of a curve defined by parametric equations \(x(t)\) and \(y(t)\) over an interval \([a, b]\) is given by the integral \(L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\). We will use this formula to find the lengths for the given parametric curves.

Step 2: Differentiate the Parametric Equations for Part (a)

For the curve (a), where \(x = \sin \theta\) and \(y = \cos \theta\), differentiate both with respect to \(\theta\):\[\frac{dx}{d\theta} = \cos \theta, \quad \frac{dy}{d\theta} = -\sin \theta\].

Step 3: Calculate the Length of the Curve for Part (a)

Substitute \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\) into the length formula: \[L = \int_0^{2\pi} \sqrt{\cos^2 \theta + (-\sin \theta)^2} \, d\theta = \int_0^{2\pi} \sqrt{\cos^2 \theta + \sin^2 \theta} \, d\theta\]. Simplify, using \(\cos^2 \theta + \sin^2 \theta = 1\): \[L = \int_0^{2\pi} 1 \, d\theta = 2\pi\].

Step 4: Differentiate the Parametric Equations for Part (b)

For the curve (b), where \(x = \sin 3\theta\) and \(y = \cos 3\theta\), differentiate both with respect to \(\theta\):\[\frac{dx}{d\theta} = 3\cos 3\theta, \quad \frac{dy}{d\theta} = -3\sin 3\theta\].

Step 5: Calculate the Length of the Curve for Part (b)

Substitute \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\) into the length formula: \[L = \int_0^{2\pi} \sqrt{(3\cos 3\theta)^2 + (-3\sin 3\theta)^2} \, d\theta = \int_0^{2\pi} \sqrt{9\cos^2 3\theta + 9\sin^2 3\theta} \, d\theta\]. Simplify using \(\cos^2 3\theta + \sin^2 3\theta = 1\): \[L = \int_0^{2\pi} 3 \, d\theta = 6\pi\].

Step 6: Explain the Difference

The lengths differ because in part (a), the curve is traced once around the unit circle as \(\theta\) goes from 0 to \(2\pi\). In part (b), the curve is traced thrice around the circle since \(3\theta\) means the parameter sweeps around the circle three times faster, leading to an increased length.

Key Concepts

Curve Length

Calculating the length of a curve described by parametric equations involves using a special formula. When given equations in the form of parametric equations \(x(t)\) and \(y(t)\), the length \(L\) of the curve over the interval \([a, b]\) can be determined using:
\[ L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \ dt \]
This formula takes into account the rate of change in both the \(x\) and \(y\) directions. By finding the derivative of each parametric equation with respect to \(t\) and integrating over the interval, you can find the curve's length.

• First, compute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\).
• Then substitute these derivatives into the length formula.
• Finally, integrate over the given interval to find the curve's total length.

The integration gives you the length by effectively "adding up" infinitely small segments of the curve's path.

Unit Circle

The unit circle is a fundamental concept in trigonometry and represents a circle with a radius of one unit. It's typically centered at the origin in a Cartesian coordinate system. Parametric equations like \(x = \sin \theta\) and \(y = \cos \theta\) describe movements along the unit circle.
For curves described with these equations, as \(\theta\) varies from 0 to \(2\pi\), the entire unit circle is traced exactly once.

• \(x = \sin \theta\) describes the horizontal position along the circle.
• \(y = \cos \theta\) describes the vertical position.

Understanding movements along the unit circle helps with visualizing how parametric curves behave, particularly how they trace out a shape like a circle. This knowledge allows us to predict and calculate the properties of these curves, such as the length, as demonstrated in the problems.

Differentiation

Differentiation is a calculus tool used to find the rate at which a function is changing. In the context of parametric equations, you differentiate each component of the curve separately.
For example, for the equation \(x=\sin \theta\), differentiating with respect to \(\theta\) gives \(\frac{dx}{d\theta} = \cos \theta\). Similarly, for \(y=\cos \theta\), the derivative is \(\frac{dy}{d\theta} = -\sin \theta\).

• These derivatives are crucial for calculating the curve's length using the curve length formula.
• They help us understand how the position along the curve changes as \(\theta\) increases.

Differentiating parametric equations is a critical step before setting up the integral needed to find the length. This processing helps prepare the functions for the next key step - integration.

Integral Calculus

Integral calculus is the branch of calculus that deals with the accumulation of quantities, such as areas under a curve or, in this case, the length of a curve. Once you have differentiated your parametric equations and substituted these into the length formula, integration is used to compute the total length.
The integral gathers all the small changes (or differential lengths) along the curve's path:

• It essentially adds up each little segment calculated from the derivatives of the parametric equations.
• The result gives you the comprehensive length of the curve between specified parameters, \(\theta\) from 0 to \(2\pi\) in this example.

For the given problems, the length of the curve was found through integrating the formula from the previously computed derivatives. Understanding the role of integration helps in grasping how we transition from the instantaneous rate of change (given by differentiation) to the total accumulation (given by integration). This complete view allows students to appreciate how intricate calculations like curve lengths are drawn from mathematical principles.