##### Find the area enclosed by the curve x = 3 cost, y = 2 sint

Given equations are x = 3 cost, y = 2 sint

These are the parametric equation of the eclipse.

Eliminating the parameter t, we get  Squaring and adding equation (i) and (ii), we get (as sin2t + cos2t = 1)

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

(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) (As x is between and the value of y varies, here y is Cartesian equation of the eclipse) (as )  Substitute So the above equation becomes,  We know, So the above equation becomes,  Apply reduction formula: On integrating we get,  Undo the substituting, we get  On applying the limits we get,  Hence the area enclosed by the curve x = 3 cost, y = 2 sint is equal to square units.

29