Question

Find the area between the ellipse
X= 3Cost , y=2sint and the
Circle X= Cost, Y=sint using
Green's.​

Answer 1

Answer:x = 3cost

=> x/3 = cost ........1

y = 2sint

=> y/2 = sint .........2

Square equation 1 and 2 and add the , we get

x2 /9 + y2 / 4 = sin2 t + cos2 t

=> x2 /9 + y2 / 4 = 1   (since sin2 t + cos2 t = 1 )

It is equation of an ellipse. So curve forms an ellipse.

Now area of ellipse = 4 *0∫3  (2/3) *√(9 - x2 ) dx

                            = (8/3) [(x/2) *√(9 - x2 ) + (9/2) * sin-1 (x/3) 0]3   (Using formula of √(a2 - x2 ) )

                            = 6π