Q. 294.3( 3 Votes )

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

Answer :

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: -


9.PNG


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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