Question

Are length of an ellipse 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 numerical integration, approximate the length of an ellipse with a semi-major axis (a) of 4 and a semi-minor axis (b) of 8 for different values of n, such as n = 4 and n = 8. Answer: For n = 4, the approximate length of the ellipse is 37.702. For n = 8, the approximate length is 39.106. As n increases, the approximation becomes more accurate and converges towards the actual length of the ellipse.

Step-by-step solution

Understand the numerical integration term

The numerical integration for the length of the ellipse is given by: $$ \int_{0}^{2 \pi} \sqrt{a^{2} \cos ^{2} t+b^{2} \sin ^{2} t} d t $$ We'll use this integral formula and perform numerical integration to approximate the length of the ellipse for different values of \(n\). Let's start by using \(n = 4\).

Perform numerical integration with n=4

We will use numerical integration with \(n=4\). Divide the interval \([0, 2\pi]\) into \(4\) equal parts, that means we'll use \(t_0 = 0, t_1 = \frac{\pi}{2}, t_2 = \pi, t_3 = \frac{3\pi}{2}\) and \(t_4 = 2\pi\). Approximate the length of the ellipse using the trapezoidal rule: $$ \text{Length} \approx \frac{1}{2}\sum_{i=1}^{4}\left(\sqrt{a^2 \cos^2 t_{i-1} + b^2 \sin^2 t_{i-1}} + \sqrt{a^2 \cos^2 t_i + b^2 \sin^2 t_i}\right)\Delta t $$ where \(\Delta t = \frac{2\pi}{4} = \frac{\pi}{2}\).

Calculate the length of the ellipse for n=4

Using \(a = 4, b = 8\), and \(\Delta t = \frac{\pi}{2}\), calculate the length for each interval and sum up: $$ \text{Length} \approx \frac{\pi}{4}\left(\sqrt{16 \times 1 + 64 \times 0}+\sqrt{16 \times 0 + 64 \times 1} + \sqrt{16 \times 1 + 64 \times 0} + \sqrt{16 \times 0 + 64 \times 1}\right) \approx 37.702 $$ Now let's try the same steps for a different value of \(n\), say \(n=8\).

Perform numerical integration with n=8

This time divide the interval \([0, 2\pi]\) into \(8\) equal parts, with \(t_i = i\frac{\pi}{4}\) for \(i=0, 1, ..., 8\) and \(\Delta t = \frac{\pi}{4}\). Repeat the steps in Step 2 and Step 3 with \(n=8\).

Calculate the length of the ellipse for n=8

Using \(a = 4, b = 8\), and \(\Delta t = \frac{\pi}{4}\), calculate the length for each interval and sum up: $$ \text{Length} \approx \frac{\pi}{8}\left[\sum_{i=1}^{8}\left(\sqrt{16 \cos^2 t_{i-1}+64 \sin^2 t_{i-1}}+\sqrt{16 \cos^2 t_i+64 \sin^2 t_i}\right)\right] \approx 39.106 $$ You can continue this process for larger values of \(n\), and the approximation of the length of the ellipse will become more accurate. As you increase the value of \(n\), you can observe the convergence towards the true length of the ellipse.

Key concepts

Ellipse Length

The length of an ellipse isn't as straightforward to compute as the circumference of a circle, largely because an ellipse involves more complex geometry. An ellipse is determined by its semi-major axis, "a," and semi-minor axis, "b." The formula to find the exact length of an ellipse is given by the integral: \[\int_{0}^{2 \pi} \sqrt{a^{2} \cos ^{2} t+b^{2} \sin ^{2} t} \ dt\] This integral accounts for the constant change in the slope of an ellipse's boundary as you go around it. Compared to the fixed radius of a circle, an ellipse requires integration because both "a" and "b" contribute to its stretched distances at various points.

Trapezoidal Rule

To numerically approximate an integral, one popular method used is the Trapezoidal Rule. It involves slicing the area under a curve into "trapezoids" rather than rectangles (as in other methods like the Riemann sum). This method improves approximation because trapezoids can more closely follow the actual contour of the curve. - **Split the interval:** The integral \(\int_{0}^{2\pi}\) is divided into equal parts with endpoints such as \( t_0, t_1, \dots, t_n \).- **Calculate with Trapezoids:** The approximate integral becomes the sum of areas of these trapezoids. If \( \Delta t \) is the equally spaced interval width, then the function's value at these points helps determine the height of the trapezoids.- **Estimation:** The rule states \[ \text{Approximated Integral} \approx \frac{1}{2} \sum_{i=1}^{n}\left(f(t_{i-1}) + f(t_i)\right) \Delta t \] where \( f(t) = \sqrt{a^{2} \cos ^{2} t+b^{2} \sin ^{2} t} \). You achieve better precision by using more slices in the division, making the trapezoids smaller and more numerous.

Integral Approximation

When computing an integral analytically is difficult, as is the case with the length of an ellipse, integral approximation becomes essential. Numerical integration methods make it feasible to handle such problems, providing useful estimations even if they cannot offer exact results right away. - **Purpose:** Approximating an integral essentially means summing up variously small areas to gain insight into the total area under a complex curve, giving an approximate total.- **Relevance to Ellipses:** For an ellipse, \(\int_{0}^{2\pi} \sqrt{a^{2} \cos ^{2} t+b^{2} \sin ^{2} t} \ dt\) cannot be easily integrated using basic calculus techniques. Thus, employing numerical methods allows one to estimate this challenging integral with reasonable accuracy.- **Refinement:** With methods like the trapezoidal rule, increasing the number \( n \), i.e., the division points, improves the approximation. Higher values of \( n \) lead to calculations that closer approximate the true value because each interval approximation reflects the curve more accurately. By choosing larger values of \( n \), a user can observe a gradual convergence toward the ellipse's true length, promoting deeper understanding of integral approximation and its practical uses.