Answer :

Given equations are:

...... (1)


And x = ae, x = 0 ...... (2)


equation (1) represents an eclipse that is symmetrical about the x - axis and also about the y - axis, with center at origin and passes through (±a, 0) and (0, ±a).


A rough sketch is given as below: -


11.PNG


We have to find the area of shaded region.


Required area


= shaded region ABCDA


= 2 (shaded region OABO) ( as it is symmetrical about the x - axis)


(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,ae) 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 bounded by the ellipse and the ordinates x = ae and x = 0, where b2 = a2 (1 - e2) and e<1 is equal to square units.


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