Question

The length of the ellipse \(x=a \cos (t), y=b \sin (t), 0 \leq t \leq 2 \pi\) is given by \(L=4 a \int_{0}^{\pi / 2} \sqrt{1-e^{2} \cos ^{2}(t)} d t\), where \(e\) is the eccentricity of the ellipse. Use Simpson's rule with \(n=6\) subdivisions to estimate the length of the ellipse when \(a=2\) and \(e=1 / 3\).

Short answer

The estimated length of the ellipse is approximately 12.57 units.

Step-by-step solution

Understand the Problem

We are given the formula for the length of an ellipse and asked to approximate it using Simpson's rule with 6 subdivisions. We have specific values for the semi-major axis \(a\) and eccentricity \(e\).

Calculate Simpson's Rule Parameters

The interval \([0, \pi/2]\) needs to be divided into 6 subdivisions, so the width of each subdivision is \(h = \frac{\pi / 2 - 0}{6} = \frac{\pi}{12}\). We then have points \(t_0, t_1, ..., t_6\) at \(0, \frac{\pi}{12}, \frac{2\pi}{12}, ..., \frac{6\pi}{12}\).

Define the Function to Integrate

The function we need to integrate from \(0\) to \(\pi/2\) is \(f(t) = \sqrt{1 - \left(\frac{1}{3}\right)^2 \cos^2(t)}\). This will be substituted in Simpson's rule formula.

Apply Simpson's Rule

Using Simpson's Rule: \(\int_{a}^{b} f(x) \, dx \approx \frac{h}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + 4f(x_5) + f(x_6)]\). Substitute \(f(t)\) and calculations for \(t_0 = 0\), \(t_6 = \frac{\pi}{2}\), with their respective multiples of coefficients.

Compute Each Function Value

Calculate each value of \(f(t)\) for \(t_0 = 0\) to \(t_6 = \frac{\pi}{2}\). For example, \(f(0) = \sqrt{1 - (1/3)^2 \, \cos^2(0)}\) simplifies to \(\sqrt{1 - 1/9}\). Compute similarly for each point.

Calculate Final Integral Approximation

Plug in all calculated \(f(t_i)\) into the Simpson's Rule formula to find the approximation. Calculate \(L = 4 \times 2 \times \text{(Simpson's Rule Result)}\).

Key concepts

Arc Length

The concept of arc length is pivotal when dealing with curves, as it essentially relates to the total distance traveled along a curved path. Calculating this distance exactly can often be complex, especially for functions that do not have simple geometric interpretations, like ellipses.
To find the arc length of a curve defined by parametric equations such as for an ellipse, calculus provides us with a formula that integrates the square root of the sum of squares of derivatives, making it possible to deal with non-linear paths.
In mathematics, the arc length formula for parametric equations \( (x(t), y(t)) \) over an interval \( [a, b] \) is given by:\[ L = \int_a^b \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 } \, dt \] This formula helps in calculating the accurate measure of curves, such as that of an ellipse, which is not straightforward due to its non-circular nature.

• Recognizes that complicated paths cannot rely on simple trigonometry.
• Requires integration over the desired interval for accuracy.

Ellipse

An ellipse is a fascinating shape that can be thought of as a stretched circle, defined mathematically as a set of all points where the sum of the distances to two fixed points, called foci, is a constant. Unlike circles, ellipses have two differing axes: the major (longer) and the minor (shorter) axis.
In coordinate geometry, an ellipse can be represented by the equation \( x = a \cos(t) \), \( y = b \sin(t) \), where \(a\) and \(b\) are the semi-major and semi-minor axes, respectively.
The total length around the ellipse or arc length, however, does not have a simple algebraic expression like that of a circle’s circumference. Instead, it requires advanced calculus and numerical approximation methods, like the one you're likely using in your exercise, to estimate.

• Histories stretch from ancient astronomy to modern physics.
• Applications are found in satellite orbits and optics.

Numerical Integration

Numerical integration is a crucial concept when it comes to calculating the integral of a function when finding an exact solution is difficult or impossible. In simple terms, numerical integration allows us to estimate the area under a curve, even if the integral cannot be expressed in a simple closed form.
One powerful numerical approximation method is Simpson’s Rule, which uses parabolas to estimate segments of the area under a curve. By dividing a region into an even number \(n\) of subintervals, Simpson’s Rule provides an excellent approximation compared to other methods like the Trapezoidal Rule.
Simpson's Rule formula for integration over \([a, b]\) is:\[ \int_a^b f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + \ldots + 4f(x_{n-1}) + f(x_n) \right] \]Where \( h = (b-a)/n \) is the width of each subdivision.

• Great for handling complex integrals that resist symbolic solutions.
• Requires equally spaced subdivisions of the total integration interval.

Eccentricity

Eccentricity is a fundamental parameter that determines the shape of an ellipse. It measures the deviation of the ellipse from being a perfect circle. This value ranges from 0 to just under 1 for ellipses, where 0 indicates a perfect circle and values closer to 1 indicate more elongated shapes.
The eccentricity \( e \) of an ellipse is defined as:\[ e = \sqrt{1 - \left(\frac{b^2}{a^2}\right)} \]where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.
Eccentricity plays a crucial role in determining the properties of the ellipse, including its arc length. It helps in understanding the geometry of an ellipse, influencing aspects like how stretched it is.

• A crucial part of understanding planetary orbits, which are elliptical.
• Difference in the lengths of axes can affect optical properties.