Find the area enclosed by the curve Y = 2 – x^{2} and the straight line x + y = 0.

To find region enclosed by

Y = 2 – x^{2} ...(i)

And x + y = 0 ...(ii)

On solving the equation (i) and (ii),

x – x^{2} + 2 = 0

Or, x^{2} – 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 – x^{2} and the straight line y + x = 0 is

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