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प्रश्न
Draw a rough sketch and find the area bounded by the curve x2 = y and x + y = 2.
उत्तर
The given curves are: x2 = y
Which is an upward parabola with vertex at origin
And line x + y = 2 ⇒ y = 2 – x
x2 = 2 – x
⇒ x2 + 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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