Book: RD Sharma - Mathematics (Volume 2)
Chapter: 21. Areas of Bounded Regions
Subject: Maths - Class 12th
Q. No. 29 of Exercise 21.1
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Find the area enclosed by the curve x = 3 cost, y = 2 sint
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: -
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.
