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


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