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 sin^{2}t + cos^{2}t = 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

= (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)

(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.

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