Question

Let g(x) = cosx², f(x) = `sqrt(x)`, and α, β (α < β) be the roots of the quadratic equation 18x² – 9πx + π² = 0. Then the area (in sq. units) bounded by the curve y = (gof)(x) and the lines x = α, x = β and y = 0, is ______.

Options

• `1/2(sqrt(3) + 1)`

• `1/2(sqrt(3) - sqrt(2))`

• `1/2(sqrt(2) - 1)`

• `1/2(sqrt(3) - 1)`

Answer 1

Let g(x) = cosx², f(x) = `sqrt(x)`, and α, β (α < β) be the roots of the quadratic equation 18x² – 9πx + π² = 0. Then the area (in sq. units) bounded by the curve y = (gof)(x) and the lines x = α, x = β and y = 0, is `underlinebb(1/2(sqrt(3) - 1))`.

Explanation:

Here, 18x² – 9πx + π² = 0

`\implies` (3x – π) (6x – π) = 0

`\implies` α = `π/6`, β = `π/3`

Also, gof(x) = cosx

∴ Required area = `int_(π//6)^(π//3) cosxdx = (sqrt(3) - 1)/2`