Find the area bounded by the ellipse and the ordinates x = ae and x = 0, where b^{2} = a^{2} (1 - e^{2}) and e<1.

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: -

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 b^{2} = a^{2} (1 - e^{2}) and e<1 is equal to square units.

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