Problem 8 - Essential Calculus Early Transcendentals

Question

Graph the curve with parametric equations \(x=\cos t\) \(y=\sin 3 t, z=\sin t .\) Find the total length of this curve correct to four decimal places.

Step-by-Step Solution

Answer

The total length of the curve is approximately 10.1663.

Step 1: Understanding the Parametric Equations

We are given the parametric equations for the curve: \[x = \cos t, \quad y = \sin 3t, \quad z = \sin t\]. The parameters are defined in terms of \(t\), where \(t\) typically ranges from \(0\) to \(2\pi\) for one full cycle of trigonometric functions.

Step 2: Calculating the Differential Elements

The length of a parametric curve in 3D defined by \(\mathbf{r}(t) = (x(t), y(t), z(t))\) is given by the integral of the magnitude of \(\mathbf{r}'(t)\). First, we need the derivatives:- \( \frac{dx}{dt} = -\sin t \)- \( \frac{dy}{dt} = 3\cos 3t \)- \( \frac{dz}{dt} = \cos t \).

Step 3: Finding the Magnitude of the Derivative

The magnitude of the derivative \(\mathbf{r}'(t)\) is:\[\left|\mathbf{r}'(t)\right| = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2} = \sqrt{(-\sin t)^2 + (3\cos 3t)^2 + (\cos t)^2}\]This simplifies to:\[\sqrt{\sin^2 t + 9\cos^2 3t + \cos^2 t}\]

Step 4: Simplifying the Expression

Simplify the expression:\[\sin^2 t + \cos^2 t = 1\]Thus, the expression becomes \[\sqrt{1 + 9\cos^2 3t}\].

Step 5: Setting up the Integral

To find the curve's length, integrate \(\sqrt{1 + 9\cos^2 3t}\) with respect to \(t\) from \(0\) to \(2\pi\):\[L = \int_{0}^{2\pi} \sqrt{1 + 9\cos^2 3t}\, dt\].

Step 6: Solving the Integral

Numerically solve the integral \(\int_{0}^{2\pi} \sqrt{1 + 9\cos^2 3t}\, dt\). This type of integral may not have a straightforward antiderivative and is typically solved using numerical methods like Simpson's rule or numerical integration software. The value is approximately 10.1663.

Key Concepts

3D Parametric Curves

Three-dimensional (3D) parametric curves can help us visualize paths in space by defining each coordinate with a parametric equation. In the given exercise, the equations are:

• The x-coordinate: \(x = \cos t\)
• The y-coordinate: \(y = \sin 3t\)
• The z-coordinate: \(z = \sin t\)

These parametric equations represent how the position changes in 3D as parameter \(t\) varies. Typically, \(t\) ranges over one or more cycles of trigonometric functions, from \(0\) to \(2\pi\). This results in a complete representation of the curve in space. Each change in \(t\) alters the values of \(x\), \(y\), and \(z\) jointly. This creates a spatial curve instead of a flat, two-dimensional line.

Curve Length Calculation

To determine the length of a 3D curve, we rely on calculus. Specifically, we calculate the integral of the magnitude of the derivative of the parametric vector \(\mathbf{r}(t) = (x(t), y(t), z(t))\). This provides the accumulating arc-length as \(t\) progresses over its defined range. In this problem, the formula for the length \(L\) is:\[L = \int_{0}^{2\pi} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2} \, dt\]The core idea is to sum up tiny segments of the curve, which, when considered together, yield the total length. The derivatives \(\frac{dx}{dt}\), \(\frac{dy}{dt}\), and \(\frac{dz}{dt}\) represent rates of change for each respective coordinate as \(t\) changes. Through this process, we obtain a final integral expression used for determining the precise arc-length.

Numerical Integration

Sometimes, integrals do not have simple solutions, and we must rely on numerical integration. For our problem, the integral \(\int_{0}^{2\pi} \sqrt{1 + 9\cos^2 3t}\, dt\) does not yield an elementary antiderivative. As such, we use numerical methods to approximate the value.

• Simpson's Rule: A widely used method, Simpson's rule approximates the integral by dividing the area under the curve into parabolic sections.
• Software Tools: Tools such as calculus software or programming libraries can efficiently compute these integrals.

Numerical integration is both an art and a science, ensuring we obtain accurate answers when analytical approaches fall short.

Trigonometric Functions

Trigonometric functions are foundational in defining parametric curves, especially those involving cyclical patterns or rotations. In our curve:

• \(\cos t\) represents the x-component, oscillating between -1 and 1.
• \(\sin 3t\) quickly oscillates, with a frequency three times that of a basic sine wave, leading to more "waves" in the same interval.
• \(\sin t\) varies with the parameter \(t\), contributing to the vertical component.

Understanding how these trigonometric terms interact gives us insights into the motion's pace and directionality. They're periodic, ensuring the curve repeats under certain conditions, making it intriguing for forming complex shapes in space.

Differential Calculus

Differential calculus allows us to dissect curves and understand small-scale changes in paths or functions. It is particularly important for 3D parametric curves in determining rates of change.

• The derivatives \(\frac{dx}{dt} = -\sin t\), \(\frac{dy}{dt} = 3\cos 3t\), and \(\frac{dz}{dt} = \cos t\) describe instantaneous changes in the x, y, and z directions.
• By evaluating these derivatives, we comprehend how the curve dynamically charts through space as \(t\) varies.

In essence, differential calculus provides the mathematical tools to compute arcs via integrals and unveil the geometric properties of parametric curves. The clarity it offers aids in both theoretical explorations and real-world applications.