The area of the loop of the folium of Dencartes x^3+y^3-3 x y=0 is (a) π(b) π/ 2 (c) 1.5 (d) ≤

Name: Problem 20, Chapter 6: Higher Engineering Mathematics
Uploaded: 2022-04-11T05:21:42-08:00
Duration: 9 min 27 s
Channel: Jeff Vermeire
Description: The area of the loop of the folium of Dencartes x^3+y^3-3 x y=0 is (a) π(b) π/ 2 (c) 1.5 (d) ≤

The area of the loop of the folium of Dencartes x^3+y^3-3 x...

Key Concepts

Area Calculation via Integration

Calculating the area enclosed by a curve typically involves setting up and evaluating a definite integral. This may be done in Cartesian, parametric, or polar coordinates depending on which representation allows for easier computation. Mastery of integral calculus and the appropriate substitution techniques is essential for computing the area of loops and other regions bounded by algebraic curves.

Parametric Representation

Parametric representation involves expressing the coordinates of points on a curve in terms of one or more parameters. This method simplifies the analysis and integration of curves, particularly when dealing with complex or implicitly defined curves. It is especially valuable for deriving equations for curves like the folium of Descartes, where direct integration can be challenging.

Polar Coordinates

Using polar coordinates is a common transformation when dealing with curves that exhibit radial symmetry or when the integration becomes simpler in terms of angles and radii. Transforming an equation from Cartesian to polar form can simplify the process of computing areas, especially for loops or curves defined implicitly.

Algebraic Curves

Algebraic curves are defined by polynomial equations in two variables. They serve as fundamental objects in algebraic geometry and provide a framework for studying the behavior, intersections, and singularities of curves in the plane. Understanding their properties is essential for analyzing their geometry and computing features such as areas enclosed by the curves.

Folium of Descartes

The folium of Descartes is a classical example of an algebraic curve defined by a cubic equation. It is notable for its distinctive loop which has important geometrical and analytical properties. Studying this curve involves exploring its singularities and the behavior of its branches, making it a useful example in the study of plane curves and implicit functions.

Transcript

00:01 So our goal is to find this shaded area here, which is called the loop in this curve.

00:09 The curve itself is called the folium of descartes.

00:15 Now, i want to do that.

00:17 The first thing we have to do is make a slight parameterization.

00:21 Y equal x times t.

00:24 We then plug that back into our original equation to get x cubed plus x cubed, t cubed equals 3 x squared t we can simplify that further and factor out that x cubed it's x cubed 1 plus t cubed equals 3 x squared t now we can solve this for x by dividing both sides by x squared 1 plus t cubed.

01:16 This gives us x equals 3t over 1 plus t cubed.

01:26 Notice that the value for y is just x times t.

01:31 So that means that y is 3t squared over 1 plus t cubed.

01:42 And one other thing, if i divide y over x over x, i get 3t squared over 1 plus t cubed times 1 plus t cubed over 3t, which simplifies the t.

02:02 So t equals y over x.

02:06 I'll change colors here.

02:09 The next step is to differentiate this entire function with respect to t.

02:15 So we get d t over d t equals now we have a quotient rule so d y that's supposed to be y d t and x x y d x d x squared d t over d t over d t is one so i have d y and x d x y over d t b t that should be an x squared x squared and multiply both sides by x squared and we get x squared equals d y x b t x d x b t and d x y over d t now i'll change color again so it's follow along we take this x squared value and we plug in our x equals 3t over 1 plus t cubed value and that gives us 9 t squared over 1 plus t squared squared multiplied both sides by d t and equals x d y minus y d x x x x x x x minus y d x x x x x x x x x x x x x x x x x x x x x x x x x x x x x...

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