Given the boundaries of the area O befound are,

• Curve is y = 2x – x²

• Line y = -x

Consider the curve

y = 2x – x²

x² -2x = - y

adding 1 on both sides

x² – 2x +1 = -(y-1)

(x-1)² = -(y-1)

This clearly shows, the curve is a parabola with vertex B (1,1)

Consider the curve, y = 2x – x² and substitute the line -x = y in the curve

-x = 2x – x²

x² – 2x – x = 0

x² – 3x = 0

x(x-3) = 0

x = 3 (or) x = 0

substituting x in y = -x

y = -3 (or) y = 0

So , the parabola meets the line y = -x at 2 points, A (3,-3) and •(0,0)

As per the given boundaries,

• The parabola y = 2x - x², with vertex at B(1,1).

• Line y = -x

The boundaries of the region to be found are,

•Point A, where the curve y = 2x - x² and the line y = -x meet i.e. A (3,-3)

•Point B, where the curve y = 2x - x² has the extreme end the vertex i.e. B (1,1)

•Point C, where the curve y = 2x - x² and the line y= -x meet i.e. C (2,0)

•Point O, the origin

Area of the required region = Area of OACBO

Area of OACBO= Area under OBCA – Area under line OA

[Using the formula ]

The Area of the required region