(b) The equation of given curve is y = 2x - x2 ⇒ x2 - 2 x = - y ⇒ x2 - 2 x + 1= - y + 1 ⇒ (x−1)2 = - ( y - 1 ) Which is the equation of parabola whose vertex is (1,1) and it is open downward.
For intersection of theparabola with the X-axis, put y = 0, then we get 0 = 2x- x2 ⇒ x(2 - x) = 0 ⇒ x = 0 ,2 Hence, area of bounded region between the curve and X-axis =