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Question
Find the area enclosed by the curve y = –x2 and the straight lilne x + y + 2 = 0
Solution
We are given that y = –x2 or x2 = –y
And the line x + y + 2 = 0
Solving the two equations,
We get x – x2 + 2 = 0
⇒ x2 – x – 2 = 0
⇒ x2 – 2x + x – 2 = 0
⇒ x(x – 2) + 1(x – 2) = 0
⇒ (x – 2)(x + 1) = 0
∴ x = –1, 2
Area of the required shaded region
= `|int_-1^2 (-x - 2) "d"x - int_-1^2 - x^2 "d"x|`
⇒ `|-[x^2/2 + x]_-1^2 + 1/3 [x^3]_-1^2|`
⇒ `|-[4/2 + 4) - (1/2 - 2)] + 1/3(8 + 1)|`
⇒ `|-(6 + 3/2) + 1/3(9)|`
⇒ `|- 15/2 + 3|`
⇒ `|(-15 + 6)/2| = |(-9)/2|`
= `9/2` sq.units
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