SOLUTION: Hi There For my maths assignment I need to prove that the area of an ellipse is given by the formula A = pi * a * b Can someone please help me? Thank you

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Question 272208: Hi There
For my maths assignment I need to prove that the area of an ellipse is given by the formula A = pi * a * b
Can someone please help me?
Thank you
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Hi There
For my maths assignment I need to prove that the area of an ellipse is given by the formula A+=+pi%2Aab 



The equation of the above ellipse is 

x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2=1

We will just find the area of the part in the first quadrant,
and then by symmetry multiply by 4 to get the area in all four 
quadrants.  We will use:

A=4int%28y%2C+dx%2C+0%2C+a+%29


We solve for y:

x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2=1

Multiply through by a%5E2b%5E2

b%5E2x%5E2%2Ba%5E2y%5E2=a%5E2b%5E2

a%5E2y%5E2=a%5E2b%5E2-b%5E2x%5E2

a%5E2y%5E2=b%5E2%28a%5E2-x%5E2%29

Take positive square roots of both sides:

ay=b%2Asqrt%28a%5E2-x%5E2%29

y=%28b%2Fa%29%2Asqrt%28a%5E2-x%5E2%29



We use trigonometric substitution:
sqrt(a^2-x^2)





-----------------------------------


x%2Fa=sin%28%22@%22%29

x=a%2Asin%28%22@%22%29

dx=a%2Acos%28%22@%22%29%2A%22d@%22

sqrt%28a%5E2-x%5E2%29%2Fa=cos%28%22@%22%29

sqrt%28a%5E2-x%5E2%29=a%2Acos%28%22@%22%29

To substitute we eill also change the limits from
limits on x to limits on @:

When x=0, %22@%22=0 and
when x=a, %22@%22=pi%2F2 


Substituting

A=4int%28+%28b%2Fa%29%2Asqrt%28a%5E2-x%5E2%29%2C+dx%2C+0%2C+a%29+ 








  




A=+2ab%28pi%2F2+%2B+%281%2F2%29%28sin%28pi%29-sin%280%29%29%29+

A=+2ab%28pi%2F2+%2B+%281%2F2%29%280-0%29%29+

A=+2ab%28pi%2F2+%2B+0%29+

A=+2ab%28pi%2F2%29+

A=+cross%282%29ab%28pi%2Fcross%282%29%29+

A=pi%2Aab

Edwin