The area bounded by the curve x=2-y-y^2 and the Y -axis is

Step-by-step solution

The curve x = 2 - y - y^2 intersects the Y-axis when x = 0 , giving y^2 + y - 2 = 0 , which factors as (y + 2)(y - 1) = 0 . The roots y = -2 and y = 1 determine the limits of integration. Between these roots, x = 2 - y - y^2 remains positive, so the area is given directly by A = _ -2 ¹ (2 - y - y^2) , dy Integrating term by term yields [2y - y^2 2 - y^3 3 ]_ -2 ¹ Evaluating at the bounds gives (2 - 1 2 - 1 3 ) - (-4 - 4 2 + 8 3 ) = 7 6 - (- 10 3 ) = 7 6 + 20 6 = 27 6 = 9 2 The area bounded by the curve and the Y-ax