Blog Entry #104. (I thought I use a different font this time. I like Arial, but change is good, at least every once in a while. )
Objective of this blog entry: calculate the area of an ellipse (Fig. A), and then a section of an ellipse (Fig. B).
Normally, the equation given for an ellipse is:
x^2/a^2 + y^2/b^2 = 1
I am going to use the parametric form
x = a cos t
y = b sin t
with 0 ≤ t ≤ 2 π
Let θ be the upper limit (angle) and the area is:
θ
∫ y dx =
0
θ
∫ y(t) x'(t) dt =
0
θ
∫ (b sin t) (a sin t) dt =
0
θ
∫ a b sin^2 t dt
0
Note: the antiderviatve of sin^2 t is 1/2 ( t  sin t cos t ).
Then:
θ
∫ a b sin^2 t dt =
0
a b [ 1/2 ( θ  sin θ cos θ )  1/2 ( 0  sin 0 cos 0 ) ] =
1/2 a b ( θ  sin θ cos θ )
The area of an ellipse (up to angle θ , see Fig. B) is:
1/2 a b ( θ  sin θ cos θ )
To find the area of the entire ellipse, let θ = 2 π. Then:
A = 1/2 a b (2 π  sin (2 π) cos (2 π)) = π a b
That concludes is blog entry. Before I go, and before I forget: THANK YOU U.S. TROOPS FOR ALL YOU DO!!!!
Eddie
A blog is that is all about mathematics and calculators, two of my passions in life.
Wednesday, May 23, 2012
Area of an Ellipse
Subscribe to:
Post Comments (Atom)
Fun with the FX603P Emulator
Fun with the FX603P Emulator Author for the Emulator: Martin Krischik Link to Emulator (Android): https://play.goo...

Casio fx991EX Classwiz Review Casio FX991EX The next incarnation of the fx991 line of Casio calculators is the fx991 EX. ...

One of the missing features of the TI82/83/84 family is the ability to convert between bases. Here are two programs in TIBasic to help...

HP Prime: Basic CAS Commands for Polynomials and Rational Expressions Define the following variables: poly: a polynomial o...
Thanks for providing such useful information about ellipse .The formulas and their derivation which are provided here are very useful.
ReplyDeleteArea of an Ellipse
You are welcome Manoj.
ReplyDelete