Book: RD Sharma - Mathematics (Volume 2)
Chapter: 21. Areas of Bounded Regions
Subject: Maths - Class 12th
Q. No. 26 of Exercise 21.1
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Find the area bounded by the ellipse 
and the ordinates x = ae and x = 0, where b2 = a2 (1 - e2) 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 b2 = a2 (1 - e2) and e<1 is equal to
square units.
