Find the area of the region bounded by the curve y=2x-x2 and the straight line y=-x.
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