Question

The length of an ellipse with axes of length \(2 a\) and \(2 b\) is $$\int_{0}^{2 \pi} \sqrt{a^{2} \cos ^{2} t+b^{2} \sin ^{2} t} d t$$ Use numerical integration and experiment with different values of \(n\) to approximate the length of an ellipse with \(a=4\) and \(b=8\)

Short answer

Question: Using the Simpson's rule for numerical integration, approximate the length of an ellipse with half major and minor axes lengths of 4 and 8, respectively. Compare the accuracy of the approximation for different values of n (number of subintervals).

Step-by-step solution

Understand the variables and equation

In this problem, we are given the formula for the length of an ellipse: $$\int_{0}^{2\pi} \sqrt{a^{2} \cos ^{2} t+b^{2} \sin ^{2} t} d t$$ Here, \(a\) and \(b\) are the lengths of the half major and minor axes of the ellipse, and \(t\) is the parameter angle that ranges from \(0\) to \(2\pi\). We will use numerical integration with different values of \(n\) to approximate the length of the ellipse given \(a=4\) and \(b=8\).

Choose an appropriate numerical integration method

One common method for numerical integration is the Simpson's rule, which divides the interval of integration into even subintervals, uses quadratic functions to approximate the integrand on each subinterval, and then sums the areas under these functions to get an approximation of the integral. We will use this method to approximate the length of the ellipse.

Implement Simpson's rule with different values of n

To implement Simpson's rule, we must choose a value for \(n\), which represents the number of subintervals. We will test different values of \(n\) to experiment with the accuracy of the approximation. Let's start with \(n=10\). After applying Simpson's rule to the formula with \(n=10\), we get an approximate value for the length of the ellipse. Calculate the approximation and record the result. Repeat the procedure for different values of \(n\). For instance, try \(n=100\), \(n=1000\), and \(n=10000\).

Compare the results for different values of n

Once you have calculated the length of the ellipse using Simpson's rule for different values of n, compare the results and analyze the convergence of the approximations. As the value of \(n\) increases, the approximation should become more accurate. Discuss the results and the effect of the choice of \(n\) on the accuracy of the approximation.

Draw a conclusion

Based on the results obtained with different values of \(n\), we can conclude that the Simpson's rule works effectively for approximating the length of an ellipse. As the value of \(n\) increases, the approximation becomes more accurate, but the computation also becomes more time-consuming. Choose a value of \(n\) that provides a satisfactory balance between accuracy and computational cost.

Key concepts

Simpson's Rule

Simpson's Rule is a numerical method used to estimate the definite integral of a function. It's especially useful when an integral cannot be solved analytically. The method approximates the area under a curve by dividing it into even subintervals and fitting parabolas to these intervals. This involves choosing an even number of subintervals, denoted as \( n \), and using points at the midpoint and endpoints of these intervals to calculate the integrals of quadratic functions.

The general idea is to improve the approximation by making these parabolas closely resemble the actual curve. The formula used in Simpson's Rule is:

\[\int_{a}^{b} f(x) \, dx \approx \frac{\Delta x}{3} \left[ f(x_0) + 4 \sum_{i=1, \text{odd}}^{n-1} f(x_i) + 2 \sum_{i=2, \text{even}}^{n-2} f(x_i) + f(x_n) \right]\]

Here, \( \Delta x = \frac{b-a}{n} \) is the width of each subinterval. By choosing different values for \( n \), we can experiment with how well the numerical method approximates the exact integral value. More subintervals generally lead to a better approximation, but also require more computation.

Ellipse Length Calculation

Calculating the length of an ellipse is not straightforward because there's no simple formula for its perimeter. Instead, one must use a more complex integral approach. The formula to compute the length around an ellipse with semi-major axis \( a \) and semi-minor axis \( b \) involves integrating the square root of a trigonometric expression:

\[L = \int_{0}^{2\pi} \sqrt{a^2 \cos^2 t + b^2 \sin^2 t} \, dt\]

In this context, \( t \) is a parameter representing the angle in radians, ranging from \( 0 \) to \( 2\pi \). While the ellipse's simple geometry suggests easier computation, these expressions can be quite complex to integrate directly.

Numerical integration methods, like Simpson's Rule, make this computation more manageable. By breaking down the entire perimeter into small segments and calculating the distance for each segment, we get an approximation for the total length.

Convergence Analysis

Convergence Analysis is an essential aspect when using numerical methods for integration, such as Simpson's Rule. Its purpose is to assess how the approximation of an integral improves as the number of subintervals \( n \) increases.

The idea is to test several values of \( n \)—such as \( n=10, 100, 1000, \) or even \( 10000 \)—to determine how the result approaches the true value. Generally, the higher the \( n \), the smaller and more accurate the segments approximating the curve. However,

• As \( n \) increases, the method typically delivers a more precise result.
• There's a trade-off as computational time and resources increase with \( n \).
• Finding a satisfactory level of accuracy involves choosing \( n \) that balances convergence with computational cost.

Understanding convergence leads to picking the most efficient \( n \), maximizing accuracy without unnecessary computation.