Book: RD Sharma - Mathematics (Volume 2)
Chapter: 21. Areas of Bounded Regions
Subject: Maths - Class 12th
Q. No. 10 of Exercise 21.1
Listen NCERT Audio Books to boost your productivity and retention power by 2X.
Draw a rough sketch of the graph of the curve 
and evaluate the area of the region under the curve and above the x - axis
Given equations are:
...... (1)
And x - axis ...... (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 (±2, 0) and (0, ±3).
A rough sketch is given as below: -
We have to find the area of shaded region.
Required area
= shaded region ABCA
= 2 (shaded region OBCO) ( 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 (0,2) 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 of the region under the given curve and above the x - axis is equal to
square units.
