Answer :

Given equation:

...... (1)



equation (1) represents a half eclipse that is symmetrical about the x - axis and also about the y - axis with center at origin and passes through (±1, 0) and (0, ±2). And x[0,1] is represented by region between y - axis and line x = 1.


A rough sketch is given as below: -


10.PNG


We have to find the area of shaded region.


Required area


= (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 (0,1) and the value of y varies)


(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 between the curve and the x - axis is equal to square units.


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