Question

Find the area enclosed by the curve y = –x² and the straight lilne x + y + 2 = 0

Answer 1

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