Question

Let f : [-1, 2] → [0, ∞] be a continuous function such that f(x) = f(1 - x) ∀ x ∈ [-1, 2].

Let R₁ = `int_-1^2 xf(x) dx` and R₂ be the area of the region bounded by y = f(x), x = -1, x = 2 and the X-axis. Then, ______

विकल्प

• R₁ = 2R₂

• R₁ = 3R₂

• 2R₁ = R₂

• 3R₁ = R₂

Answer 1

Let f : [-1, 2] → [0, ∞] be a continuous function such that f(x) = f(1 - x) ∀ x ∈ [-1, 2].

Let R₁ = `int_-1^2 xf(x) dx` and R₂ be the area of the region bounded by y = f(x), x = -1, x = 2 and the X-axis. Then, 2R₁ = R₂

Explanation:

`R_1 = int_-1^2 x f(x) dx`

= `int_-1^2(1 - x)f(1 - x)dx` .............`[∵ int_a^b f(x)dx = int_a^bf(a + b - x)dx]`

= `int_-1^2(1 - x) f(x) dx` ............[∵ f(x) = f(1 - x)(given)]

∴ `R_1 = int_-1^2 f(x) dx - R_1 ⇒ 2R_1 = int_-1^2 f(x) dx`

According to the given condition, 

`R_2 = int_-1^2 f(x) dx`

∴ R₂ = 2R₁