Find the area of the region included between the parabola y2 = x and the line x + y = 2.
To find region enclosed by
y2 = x ...(i)
And x + y = 2 ...(ii)
From equation (i) and (ii),
y2 + y – 2 = 0
Or, (y + 2) (y – 1) = 0
Or, y = – 2, 1
x = 4,1
Equation represents a parabola with vertex at origin and its axis as x - axis
Equation represents a line passing through (2,0) and (0,2)
On solving these two equations, we get point of intersections. The points of intersection of line and parabola are (1,1) and (4, – 2) These are shown in the graph below
Shaded region represents the required area. We slice it in rectangles of width Δy and length = (x1 – x2).
Area of rectangle = (x1 – x2)Δy.
Required area of Region AOBA
The area of the region included between the parabola y2 = x and the line x + y = 2.