Question

Find the area of the region enclosed by the parabola x² = y and the line y = x + 2

Answer 1

Here, x² = y and y = x + 2

∴ x² = x + 2

⇒ x² – x – 2 = 0

⇒ x² – 2x + x – 2 = 0

⇒ x(x –2) + 1(x – 2) = 0

⇒ (x – 2)(x + 1) = 0

∴ x = –1, 2

Graph of y = x + 2

x	0	–2
y	2	0

Area of the required region

= `int_(-1)^2 (x + 2)"d"x - int_(-1)^2 x^2  "d"x`

= `[x^2/2 + 2x]_(-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`

= `9/2` sq.units

Hence, the required area = `9/2` sq.units