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Draw a rough sketch and find the area bounded by the curve x^{2} = y and x + y = 2.

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#### Solution

The given curves are: x^{2} = y

Which is an upward parabola with vertex at origin

And line x + y = 2 ⇒ y = 2 – x

x^{2} = 2 – x

⇒ x^{2} + x – 2 = 0

⇒ (x + 2)(x – 1) = 0

⇒ x = -2 and x = 1

Now, y = 2-(-2) = 4

and y = 2 – 1 ⇒ y = 1

⇒ y = 4 and y = 1

Thus, the points of intersection are (-2, 4) and (1, 1)

The required area of the shaded region

`= int_-2^1 (2 - "x") "dx" - int_-2^1 "x"^2 "dx"`

`= |2"x" - "x"^2/2|_-2^1 - |"x"^3/3|_-2^1`

`= 2 - 1/2 + 4 + 4/2 - 1/3 - 8/3`

`= (12 - 3 + 24 + 12 - 2 - 16)/6`

`= 9/2` sq.units

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