Question

Find the area of the region bounded by the curves y = x² + 2, y = x, x = 0 and x = 3.

Answer 1

Given:

y = x² + 2, y = x, x = 0 and x = 3

y = x² + 2

y − 2 = x²

x² = y − 2

Point D is intersection of x = 3 and parabola putting x = 3 in x² = y − 2

⇒ 3² = (y − 2)

⇒ 9 = y − 2

⇒ y = 11

⇒ B = (3, 11)

C is the intersection of x = 3 and y = x

⇒ y = 3

⇒ C = (3, 3)

Area (POCD) =Area (POED) − Area (OEC)

Area (POED) = `∫_0^3 y  dx`

= `∫_0^3 (x^2 + 2)dx`    [∵ y = x² + 2]

= `(x^3/3 + 2x)_0^3`

= `(27/3 + 6 − 0/3)`

= 9 + 6 = 15

Area (OEC) = `∫_0^3 y  dx`

= `∫_0^3 x dx`     [∵ y = x]

= `(x^2/2)_0^3`

= `9/2`

= Area (POCD) = 15 − `9/2`

= `21/2` sq. unit