Given equations are x = 3 cos t, y = 2 sin t
These are the parametric equation of the eclipse.
Eliminating the parameter t, we get
Squaring and adding equation (i) and (ii), we get
This is Cartesian equation of the eclipse.
A rough sketch of the circle is given below: -
We have to find the area of shaded region.
Required area
= (shaded region ABCDA)
= 4(shaded region OBCO)
(the area can be found by taking a small slice in each region of width Δx, then the area of that sliced part will be yΔx as it is a rectangle and then integrating it to get the area of the whole region)
\(=4\int^3_0 ydx\) (As x is between (0, 3) and the value of y varies, here y is Cartesian equation of the eclipse)
So the above equation becomes,
So the above equation becomes,
Apply reduction formula:
Hence the area enclosed by the curve x = 3 cos t, y = 2 sin t is equal to 6π square units.
