Q26 of 43 Page 1

Find the area of the region {(x, y) : x2 + y2≤ 4, x + y ≥ 2}.

Region is defined as- x2 + y2 ≤ 4, x + y ≥ 2


x2 + y2 = 4 represents a circle with center at (0,0) and x+y=2 is line.


First, let us calculate the points of intersection,


Putting y = 2 – x in the equation of a circle, we get,


x2 + (2 – x)2 = 4


x2 + 4 + x2 – 4x = 4


2x2 – 4x = 0


2x(x – 2) = 0


x = 0 or x = 2.


So, the points of intersection are (2, 0) and (0, 2).


The region defined with inequation is shaded in the figure.


Required area = area under circle – area under the line(x + y= 2)



Note- Formula to be used:


⇒ Area =


∴ Area = 2 Sin⁻11 – 2


⇒ Area = (π – 2) sq. units


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