Let f : [–1, 2] \rightarrow [0, \infty] be a continuous function such that f (x) = f (1 – x) for all x \in [–1, 2]. Let R_{1} = \int_{- 1}^{2} x \textrm{ } f \left(\right. x \left.\right) \textrm{ } dx, and R2 be the area of the region bounded by y = f (x), x =

Engineering

Mathematics

Introduction to Area

Question

Let f : [–1, 2] $\to$ [0, $\infty$ ] be a continuous function such that f (x) = f (1 – x) for all x $\in$ [–1, 2].
Let $R_{1} = \int_{- 1}^{2} x f (x) dx$ , and R₂ be the area of the region bounded by y = f (x), x = – 1, x = 2 and the x‑axis. Then :

R₁ = 2R₂

2R₁ = R₂

R₁ = 3R₂

3R₁ = R₂

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$R_{1} = \int_{- 1}^{2} xf (x) dx$ .................(1)

$R_{1} = \int_{- 1}^{2} (1 - x) (1 - x) dx$

$= \int_{- 1}^{2} (1 - x) f (x) dx$ ................(2)

(1) + (2)

$2 R_{1} = \int_{- 1}^{2} f (x) dx = R_{2}$

$∴$ 2R₁ = R₂