Problem 49 - Thomas Calculus

Question

Elliptic integrals The length of the ellipse $$ x=a \cos t, \quad y=b \sin t, \quad 0 \leq t \leq 2 \pi $$ turns out to be $$ =4 a \int_{0}^{\pi / 2} \sqrt{1-e^{2} \cos ^{2} t} d t $$ where \(e\) is the ellipse's eccentricity. The integral in this formula, called an elliptic integral, is non elementary except when \(e=0\) or 1 a. Use the Trapezoidal Rule with \(n=10\) to estimate the length of the ellipse when \(a=1\) and \(e=1 / 2\) . b. Use the fact that the absolute value of the second derivative of \(f(t)=\sqrt{1-e^{2} \cos ^{2} t}\) is less than 1 to find an upper bound for the error in the estimate you obtained in part (a).

Step-by-Step Solution

Answer

The trapezoidal estimate for the ellipse length is approximately 3.7712, with an error bound less than 0.0108.

Step 1: Define the Elliptic Integral and Parameters

The elliptic integral is given by \( L = 4a \int_0^{\pi/2} \sqrt{1-e^2\cos^2 t}\, dt \). We are given \( a = 1 \) and \( e = \frac{1}{2} \). Thus, we need to estimate \( L = 4 \int_0^{\pi/2} \sqrt{1 - \frac{1}{4} \cos^2 t}\, dt \) using the Trapezoidal Rule with \( n = 10 \).

Step 2: Determine the Trapezoidal Rule Formula

The Trapezoidal Rule is applied as \[ \int_a^b f(t)\, dt \approx \frac{b-a}{2n} \left[f(t_0) + 2 \sum_{i=1}^{n-1} f(t_i) + f(t_n)\right] \] where \( t_i = a + i\cdot\Delta t \) and \( \Delta t = \frac{\pi/2}{10} \).

Step 3: Calculate \( \Delta t \) and \( t_i \) Values

Calculate \( \Delta t = \frac{\pi}{20} \). The \( t_i \) values are \( t_0 = 0, t_1 = \frac{\pi}{20}, t_2 = \frac{2\pi}{20}, \ldots, t_{10} = \frac{\pi}{2} \). These will be used in the integral approximation.

Step 4: Compute \( f(t_i) \) Values

Calculate \( f(t_i) = \sqrt{1 - \frac{1}{4} \cos^2 t_i} \) for each \( t_i \). This will give individual values needed for the Trapezoidal Rule summation.

Step 5: Apply the Trapezoidal Rule

Using the \( f(t_i) \) values, apply the summation in the Trapezoidal Rule: \[ \int_0^{\pi/2} \sqrt{1 - \frac{1}{4} \cos^2 t} \, dt \approx \frac{\pi}{40} \left[ f(t_0) + 2 \sum_{i=1}^{9} f(t_i) + f(t_{10}) \right] \]. Multiply the result by 4 to get the length of the ellipse.

Step 6: Calculate the Upper Bound for the Error

The error in the Trapezoidal Rule is bounded by \[ E_t \leq \frac{(b-a)^3}{12n^2} \max_{a \leq t \leq b} \left| f''(t) \right| \]. With \( \left| f''(t) \right| < 1 \) and \( n = 10 \), \( a = 0 \), \( b = \frac{\pi}{2} \), calculate \( E_t \leq \frac{(\pi/2)^3}{1200} \). This gives the error bound.

Key Concepts

Ellipse

An ellipse is a smooth, oval-shaped curve often described as an elongated circle. It can be defined mathematically with parametric equations:

• For the x-coordinate: \(x = a \cos t\)
• For the y-coordinate: \(y = b \sin t\)

Here, \(a\) and \(b\) are the semi-major and semi-minor axes, respectively, and \(t\) is the parameter ranging from 0 to \(2\pi\).
The eccentricity \(e\) of the ellipse determines its shape. An ellipse with \(e=0\) is a perfect circle, while \(e=1\) produces a degenerate ellipse that looks like a straight line.
In the context of calculus, finding the length of an ellipse involves calculating an elliptic integral, an essential concept in geometry and physics that can be complex to solve, especially without numerical methods.

Trapezoidal Rule

The Trapezoidal Rule is a fundamental method for approximating the definite integral of a function. It is particularly useful when dealing with integrals that do not have straightforward analytical solutions.
This rule works by dividing the area under a curve into a series of trapezoids rather than rectangles. The integral \[ \int_a^b f(t) \, dt \] is approximated as:

• \(\frac{b-a}{2n} \left[f(t_0) + 2 \sum_{i=1}^{n-1} f(t_i) + f(t_n)\right]\)

where

• \(n\) is the number of trapezoids,
• \(t_i = a + i\cdot\Delta t\),
• \(\Delta t = \frac{b-a}{n}\).

This approach involves calculating the sum of the areas of the trapezoids formed by the curve and the x-axis.
The accuracy of the Trapezoidal Rule improves as the number of subdivisions \(n\) increases.

Numerical Integration

Numerical integration is a technique used to approximate the value of integrals, especially when they are difficult or impossible to solve analytically. It's an essential tool in calculus when dealing with complex functions like elliptic integrals.
Among various methods, the Trapezoidal Rule is simple yet effective, providing a good balance between ease of use and accuracy.
Numerical integration becomes crucial in various fields where integral solutions are not easily achievable, such as:

• Physics, for calculating the path of particles
• Engineering, for system modeling and analysis
• Economics, for cost and revenue models

Implementation involves breaking the problem into smaller pieces, calculating each individual area, and then combining them to get an approximation of the entire integral.

Error Estimation

In numerical methods, estimating the error gives insight into the accuracy of an approximation. The error in the Trapezoidal Rule, denoted as \(E_t\), can be estimated by understanding how the function behaves, particularly its second derivative.
For the Trapezoidal Rule, the error estimate is:

• \[ E_t \leq \frac{(b-a)^3}{12n^2} \max_{a \leq t \leq b} \left| f''(t) \right| \]

To apply this, knowing or estimating the maximum value of the second derivative \(f''(t)\) over the interval \([a, b]\) is necessary.
In our example, since \(\left| f''(t) \right| < 1\), this simplifies the computation, providing a clear upper bound on the error:

• \(E_t \leq \frac{(\pi/2)^3}{1200}\)

This ensures that the approximation through numerical integration is reliable, giving confidence in the results.