Ellipse Areas
Name: Ernest Ng
Class: 4G (23)
Date: 2-7-06
Mathematics ACE: Ellipse Areas
Before we embark on solving the problem, let us first explore the definition of ellipse.
[pic]
An ellipse is a curve that is the locus of all points in the plane the sum of whose distances [pic] and [pic] from two fixed points [pic] and [pic] (the foci) separated by a distance of [pic] is a given positive constant [pic]
[pic]
While [pic] is called the major axis, [pic] is the semi major axis, which is exactly half the distance across the ellipse. Similarly, the corresponding parameter [pic] is known as the semi minor axis.
Parallels drawn from the formula for the area of circle ([pic]) and formula for the area of an ellipse (A = [pic]ab)
Formula for the area of a circle:[pic] where [pic] is the area, and [pic] is the radius.
In the case of a circle, radius a represents the semi major axis while radius b represents the semi minor axis. One can thus find the area of the circle through the formula A = [pic]ab, where a is equal to b. Hence circle, in actual fact, is a unique case of ellipse.
Proving that the area of an ellipse is πab
Procedures to take (Theory)
1. We have to let an ellipse lie along the x-axis and find the equation of the ellipse curve. 2. Upon finding the equation of the ellipse curve, we have to change the subject of the equation to y. (otherwise, we will have to do integration with respect to y if the subject of the equation is x.) 3. Next, applying what we have learnt, we can find the area bounded by the ellipse curve through definite integration between the 2 limits.
Calculations (Practical)
(Working steps are adapted from http://mathworld.wolfram.com/Ellipse.html)
Let an ellipse lie along the x-axis and find the equation of the figure where [pic]and [pic]are at [pic]and [pic].
a) Form an equation.
[pic]
b) Bring the second term to the right side and square both sides.
[pic]
Class: 4G (23)
Date: 2-7-06
Mathematics ACE: Ellipse Areas
Before we embark on solving the problem, let us first explore the definition of ellipse.
[pic]
An ellipse is a curve that is the locus of all points in the plane the sum of whose distances [pic] and [pic] from two fixed points [pic] and [pic] (the foci) separated by a distance of [pic] is a given positive constant [pic]
[pic]
While [pic] is called the major axis, [pic] is the semi major axis, which is exactly half the distance across the ellipse. Similarly, the corresponding parameter [pic] is known as the semi minor axis.
Parallels drawn from the formula for the area of circle ([pic]) and formula for the area of an ellipse (A = [pic]ab)
Formula for the area of a circle:[pic] where [pic] is the area, and [pic] is the radius.
In the case of a circle, radius a represents the semi major axis while radius b represents the semi minor axis. One can thus find the area of the circle through the formula A = [pic]ab, where a is equal to b. Hence circle, in actual fact, is a unique case of ellipse.
Proving that the area of an ellipse is πab
Procedures to take (Theory)
1. We have to let an ellipse lie along the x-axis and find the equation of the ellipse curve. 2. Upon finding the equation of the ellipse curve, we have to change the subject of the equation to y. (otherwise, we will have to do integration with respect to y if the subject of the equation is x.) 3. Next, applying what we have learnt, we can find the area bounded by the ellipse curve through definite integration between the 2 limits.
Calculations (Practical)
(Working steps are adapted from http://mathworld.wolfram.com/Ellipse.html)
Let an ellipse lie along the x-axis and find the equation of the figure where [pic]and [pic]are at [pic]and [pic].
a) Form an equation.
[pic]
b) Bring the second term to the right side and square both sides.
[pic]
References: 1. http://library.thinkquest.org/22584/temh3031.htm 2. http://mathworld.wolfram.com/Ellipse.html 3. http://epubs.siam.org/fulltext/SIREV/volume-42/35899.pdf 4. http://en.wikipedia.org/wiki/Main_Page - END - [pic][pic][pic] ----------------------- a b