Question

Find the area of the region bounded by the curve y = x³ and y = x + 6 and x = 0

Answer 1

We are given that: y = x³, y = x + 6 and x = 0

Solving y = x³ and y = x + 6

We get x + 6 = x³

⇒ x³ – x – 6 = 0

⇒ x²(x – 2) + 2x(x – 2) + 3(x – 2) = 0

⇒ (x – 2)(x² + 2x + 3) = 0

x² + 2x + 3 = 0 has no real roots.

∴ x = 2

∴ Required area of the shaded region

= `int_0^2 (x + 6) "d"x - int_0^2 x^3  "d"x`

= `[x^2/2 + 6x]_0^2 - 1/4 [x^4]_0^2`

= `(4/2 + 12) - (0 + 0) - 1/4 [(2)^4 - 0]`

= `14 - 1/4 xx 16`

= 14 – 4

= 10 sq.units