Book: RD Sharma - Mathematics (Volume 2)
Chapter: 21. Areas of Bounded Regions
Subject: Maths - Class 12th
Q. No. 27 of Exercise 21.1
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Find the area of the minor segment of the circle x2 + y2 = a2 cut off by the line 
.
Given equations are :
x2 + y2 = a2 ...... (1)
..... (2)
Equation (1) represents a circle with centre (0,0) and radius a, so it meets the axes at (±a,0), (0,±a).
Equation (2) represents a line parallel to y axis.
A rough sketch of the circle is given below: -
We have to find the area of shaded region.
Required area
= (shaded region BCDB)
= 2(shaded region ABCA) (as the circle is symmetrical about the x - axis as well as the y - 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 
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 minor segment of the circle x2 + y2 = a2 cut off by the line
is equal to
square units.
