The area of smaller part between the circle x^2+y^2=4 and the line x=1 is _______ sq.units.

Step-by-step solution

The area of the smaller segment bounded by the circle x^2 + y^2 = 4 (radius r = 2 ) and the vertical line x = 1 can be determined using integration or geometry. Using integration, the area is given by A = 2 ₁² 4 - x^2 , dx . The standard antiderivative yields: A = 2 [ x 2 4 - x^2 + 2 ⁻¹ ( x 2 ) ]₁² Evaluating from x = 1 to x = 2 gives: A = 2 [ ( 0 + 2 2 ) - ( 1 2 3 + 2 6 ) ] = 2 ( - 3 - 3 2 ) Simplifying results in A = 4 3 - 3 . Geometrically, the intersections at x = 1 occur at y = 3 , forming a central angle of 2