Problem 110 - Calculus 3

Question

Set up an integral to find the circumference of the ellipse with the equation \(\mathbf{r}(t)=\cos t \mathbf{i}+2 \sin t \mathbf{j}+0 \mathbf{k} .\)

Step-by-Step Solution

Answer

Set up integral: \( C = \int_0^{2\pi} \sqrt{1 + 3\cos^2 t} \; dt \).

Step 1: Recognize the Parametric Equation of the Ellipse

The given parametric equations for the ellipse are: \[ x(t) = \cos t, \quad y(t) = 2\sin t \]These parameters describe the locus of points on the ellipse as functions of \(t\), where \(t\) ranges over \([0, 2\pi]\).

Step 2: Identify the Components of the Velocity Vector

The velocity vector \( \mathbf{v}(t) \) is the derivative of the position vector \( \mathbf{r}(t) \) with respect to \(t\). Calculate the derivatives:\[ \frac{dx}{dt} = -\sin t, \quad \frac{dy}{dt} = 2\cos t \]Thus, the velocity vector is:\[ \mathbf{v}(t) = -\sin t \mathbf{i} + 2\cos t \mathbf{j} \]

Step 3: Calculate the Magnitude of the Velocity Vector

The magnitude of \( \mathbf{v}(t) \) gives the speed along the ellipse, given by:\[ |\mathbf{v}(t)| = \sqrt{(-\sin t)^2 + (2\cos t)^2} = \sqrt{\sin^2 t + 4\cos^2 t} \]Simplifying inside the square root using trigonometric identities:\[ = \sqrt{1 + 3\cos^2 t} \]

Step 4: Set Up the Integral for Circumference

The circumference \(C\) of the ellipse can be found by integrating the magnitude of the velocity vector over the interval \([0, 2\pi]\):\[ C = \int_0^{2\pi} \sqrt{1 + 3\cos^2 t} \; dt \]

Step 5: Comment on the Solution Method

The integral involves a trigonometric expression that cannot be solved using elementary functions. Numerical methods or elliptic integrals are usually employed to evaluate such integrals. The setup is crucial and complete for further computational evaluation.

Key Concepts

Parametric Equations

Parametric equations are a powerful way to define curves by expressing the coordinates of the points along the curve as functions of a parameter, typically denoted by \( t \). For the ellipse in our exercise, the equations are given as \( x(t) = \cos t \) and \( y(t) = 2\sin t \). By varying \( t \) from 0 to \( 2\pi \), these equations map out the continuous locus of points that form the ellipse. The parameter \( t \) typically represents an angle in trigonometric context.

• The term \( \cos t \) controls the movement along the x-axis.
• The term \( 2\sin t \) scales the motion along the y-axis, giving the ellipse its stretched shape.

Essentially, parametric equations provide a set view of how objects trace paths over time, and are very helpful for curves like ellipses where standard forms are more complex.

Velocity Vector

When dealing with parametric equations, the velocity vector \( \mathbf{v}(t) \) plays an integral role in understanding motion along the curve. Here, the velocity vector is derived by differentiating the position vector \( \mathbf{r}(t) \) with respect to \( t \). This gives us the components \( \frac{dx}{dt} = -\sin t \) and \( \frac{dy}{dt} = 2\cos t \).

• Velocity vector \( \mathbf{v}(t) \) directly shows directional movement and rate of change at each point on the ellipse.
• It's represented as \( \mathbf{v}(t) = -\sin t \mathbf{i} + 2\cos t \mathbf{j} \), giving both horizontal and vertical motion components.

Analyzing the velocity vector helps determine the speed and direction at which a point moves along the ellipse. This velocity affects how the circumference will be computed.

Integral Setup

Setting up the integral is a crucial step for finding the circumference of an ellipse using parametric equations. The integral computes the total arc length by summing up infinitesimal distances along the curve as determined by the velocity vector magnitude, \( |\mathbf{v}(t)| \). For our ellipse, this results in setting up the integral from 0 to \( 2\pi \) as:
\[C = \int_0^{2\pi} \sqrt{1 + 3\cos^2 t} \; dt\]

• Integrating over the range \([0, 2\pi]\) ensures the full traversal around the ellipse.
• The expression \( \sqrt{1 + 3\cos^2 t} \) accounts for variations in speed as the ellipse stretches more along the y-axis.

Although the integral seems complex, setting it up correctly is the foundation towards calculating or approximating the circumference using more advanced methods.

Trigonometric Identities

Trigonometric identities are mathematical equalities involving trigonometric functions that aid in simplifying expressions during calculations. In our exercise, the integral involves \( \sin^2 t \) and \( \cos^2 t \), calling for trigonometric simplification. The identity \( \sin^2 t + \cos^2 t = 1 \) directly helps reduce the expression under the square root:

• Initially, we have \( \sin^2 t + 4\cos^2 t \).
• Using \( \sin^2 t + \cos^2 t = 1 \), the expression simplifies to \( 1 + 3\cos^2 t \).

This simplification makes the integral setup manageable and aids in understanding the dynamics of the function as it applies to calculating the ellipse's circumference. With these identities, we can tackle complex mathematical expressions and aid computational efficiency.