The area (in sq. units) of the region lying in the first quadrant and enclosed by the X -axis, the straight line x- 3 y=0 and the circle x^2+y^2=4 , is

Step-by-step solution

Equation of given circle aligned & x^2+y^2=2^2 & radius =2 aligned and line x- 3 y=0 aligned ( 3 y)^2+y^2 & =4 & 4 y^2=4 y^2=1 y= 1 aligned If y=+1 x= 3 If y=-1 x=- 3 As point C is in first quadrant C is ( 3 , 1) Area of shaded region aligned & = ₀^ 3 x 3 d x+ _ 3 ^2 4-x^2 d x & = 1 3 ₀^ 3 x d x+ _ 3 ^2 2^2-x^2 d x aligned aligned & = 1 3 [ x^2 2 ]₀^ 3 + x 2 2^2-x^2 + 2^2 2 ⁻¹ x 2 ]_ 3 ^2 & [ a^2-x^2 d x= x 2 a^2-x^2 + a^2 2 ⁻¹ x a +c & = 1 3 3 2 + (0+2 ⁻¹ 1 )- ( 3 2 . . & = 3 -3 2 + [2 2 - 3 2 -2 3 & = 3 2 + - 3 2