Question

Let f : [–2, 3] `rightarrow` [0, ∞) be a continuous function such that f(1 – x) = f(x) for all x ∈ [–2, 3]. If R₁ is the numerical value of the area of the region bounded by y = f(x), x = –2, x = 3 and the axis of x and R₂ = `int_-2^3 xf(x)dx`, then ______.

Options

• 3R₁ = 2R₂

• 2R₁ = 3R₂

• R₁ = R₂

• R₁ = 2R₂

Answer 1

Let f : [–2, 3] `rightarrow` [0, ∞) be a continuous function such that f(1 – x) = f(x) for all x ∈ [–2, 3]. If R₁ is the numerical value of the area of the region bounded by y = f(x), x = –2, x = 3 and the axis of x and R₂ = `int_-2^3 xf(x)dx`, then R₁ = 2R₂.

Explanation:

We have

R₂ = `int_-2^3 xf(x)dx`

= `int_-2^3 (1 - x)f(1 - x)dx` ...`["Using" int_a^b f(x)dx = int_a^b f(a + b - x)dx]`

⇒ R₂ = `int_-2^3 (1 - x) f(x)dx` ...(∵ f(x) = f(1 – x) on [–2, 3])

∴ R₂ + R₂ = `int_-2^3 xf(x) dx + int_-2^3 (1 - x) f(x) dx`

= `int_-2^3 f(x) dx`

= R₁

`\implies` 2R₂ = R₁