Find the area enclosed by the curve Y = 2 – x2 and the straight line x + y = 0.
To find region enclosed by
Y = 2 – x2 ...(i)
And x + y = 0 ...(ii)
On solving the equation (i) and (ii),
x – x2 + 2 = 0
Or, x2 – x + 2 = 0
Or, x = 2 or – 1
∴ y = – 2 or 1
Equation (i) represents a parabola with vertex (0, 2) and downward, meets axes at (±√2, 0).
Equation (2) represents a line passing through (0, 0) and (2, – 2).
The points of intersection are A (2, – 2) and C ( – 1,1).
These are shown in the graph below: -
Required area = Region ABPCOA
The area enclosed by the curve Y = 2 – x2 and the straight line y + x = 0 is