Show that the total length of the ellipse x=a sin=b cosθ, a>b>0, is L=4 a ∫0^π/ 2 √(1-e^2 sin^2

Show that the total length of the ellipse x=a sin=b cosθ,...

Key Concepts

Parametric Representation of Curves

This concept involves expressing the coordinates of points on a curve as functions of a parameter. In the context of conic sections like an ellipse, parametric equations provide a convenient way to describe the shape and facilitate the computation of geometric properties, such as length, by allowing the application of calculus in a straightforward manner.

Arc Length of Parametric Curves

The arc length formula for a curve given in parametric form involves computing the integral of the square root of the sum of the squares of the derivatives of the coordinate functions. This method allows us to determine the total distance along a curved path and is essential when the curve is not easily represented as a single function of x or y.

Elliptic Integrals

Elliptic integrals arise in the calculation of the arc length of an ellipse, among other applications, and cannot generally be expressed in terms of elementary functions. They are integrals of functions involving square roots of polynomials of degree three or four and are categorized (as complete or incomplete) based on their limits of integration. Their use is fundamental in problems where classical integration techniques do not yield closed-form solutions.

Eccentricity of an Ellipse

Eccentricity is a measure of how much an ellipse deviates from being a circle. It is defined as the ratio of the distance between the center and a focus to the semi-major axis length. In computations involving the ellipse, the eccentricity plays an important role in modulating the shape and appears in integral expressions, such as the one for the ellipse's perimeter, influencing the form of the elliptic integrals involved.

Transcript

00:01 So in this problem, we're asked to show that the length of this ellipse is something rather specific.

00:09 So here's what we're going to do.

00:14 Well, we're just going to use our formula for the length to show this.

00:18 Now, we need to figure out what we want for our bounds, for our integral.

00:27 And what we're going to do is this is an ellipse, right? and it's symmetric.

00:42 Each of its, if you split it up into four parts, my picture doesn't look like it, but this part is symmetric to this part, we're just this symmetric to that part and is symmetric to this part.

00:55 So we really only have to measure one of its parts, say this one.

01:04 Well, then that angle would be going from 0 to pi over 2 because we only have to measure, only are measuring one of these.

01:17 Will need to multiply by 4 so that we get the total length of the ellipse.

01:27 Okay.

01:29 So with that, we can discuss the inside of our integral now.

01:39 So for the inside, we will have dx, d theta, squared, plus dyd theta.

01:59 And i will be integrating with respect to theta.

02:04 Dx d theta is really just a cosine theta.

02:12 Remember we're treating a as a constant.

02:18 And dyd theta would be negative b sine theta.

02:30 So with that, i can now try to manipulate this to get what they wanted us to have.

02:40 So this is 4, the integral from 0 to pi over 2, times the square root, a squared, cosine squared theta, minus b squared, sine squared theta, d theta.

03:02 So i know with what i'm trying to head towards, it only has a sine squared in it.

03:11 So i'm going to use a trick identity to write this cosine squared in terms of sine squared.

03:19 So that would be cosine squared theta is equal to 1 minus sine squared theta.

03:48 Oh, i'm going to drop to squared on that b.

03:55 So if i distributed that out, i would get the following.

04:07 A squared minus a squared, sine squared theta, minus b squared, sine squared, theta, d theta.

04:24 So i only have one in the answer they write.

04:28 They only have one sign squared.

04:31 So i'm going to factor that out.

04:34 So i have a squared minus...

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