The area of the region in the first quadrant enclosed by the x -axis, the line x = 3 ,y and the circle x^2 + y^2 = 4 is

Step-by-step solution

The given circle is x^2 + y^2 = 4 , which has its center at the origin (0,0) and radius r = 2 . The given line is x = 3 y , which can be rewritten as y = 1 3 x . The angle made by this line with the positive x -axis is = ⁻¹ ( 1 3 ) = 6 . The required region in the first quadrant is a sector of the circle bounded by the x -axis ( = 0 ) and the line ( = 6 ). The area of a circular sector is given by 1 2 r^2 . Area = 1 2 (2)^2 6 = 3 sq units. Alternatively, using integration by finding the intersection point ( 3 , 1)