Book: RD Sharma - Mathematics (Volume 2)
Chapter: 21. Areas of Bounded Regions
Subject: Maths - Class 12th
Q. No. 12 of Exercise 21.1
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Draw a rough sketch of the graph of the function y=2√1–x2,x[0,1] and evaluate the are enclosed between the curve and the x–axis.
Given equation:
...... (1)
equation (1) represents a half eclipse that is symmetrical about the x - axis and also about the y - axis with center at origin and passes through (±1, 0) and (0, ±2). And x∈[0,1] is represented by region between y - axis and line x = 1.
A rough sketch is given as below: -
We have to find the area of shaded region.
Required area
= (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 (0,1) 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 enclosed between the curve and the x - axis is equal to
square units.
