Book: RD Sharma - Mathematics (Volume 2)
Chapter: 21. Areas of Bounded Regions
Subject: Maths - Class 7th
Q. No. 15 of Exercise 21.1
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Using definite integrals, find the area of circle x2 + y2 = a2
Given equations are :
x2 + y2 = a2 ...... (1)
Equation (1) represents a circle with centre (0,0) and radius a, so it meets the axes at (±a,0), (0,±a). A rough sketch of the curve is given below: -
We have to find the area of shaded region.
Required area
= (shaded region ABCDA)
= 4(shaded region OBCO) (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 (0,a) 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 circle x2 + y2 = a2 is equal to
square units.
