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

Let $f:[-1,2] \rightarrow[0, \infty)$ be a continuous function such that $f(x)=f(1-x), \forall x \in[-1,2]$ If $R_{1}=\int_{-1}^{2} x f(x) d x$ and $R_{2}$ are the area of the region bounded by $y=f(x)$ $x=-1, x=2$ and the $X$ -axis. Then, [Single Correct Option 2011] (a) $R_{1}=2 R_{2}$ (b) $R_{1}=3 R_{2}$ (c) $2 R_{1}=R_{2}$ (d) $3 R_{1}=R_{2}$

Let $f:[-1,2] \rightarrow[0, \infty)$ be a continuous function such that $f(x)=f(1-x), \forall x \in[-1,2]$ If $R_{1}=\int_{-1}^{2} x f(x) d x$ and $R_{2}$
are the area of the region bounded by $y=f(x)$ $x=-1, x=2$ and the $X$ -axis. Then, [Single Correct Option 2011]
(a) $R_{1}=2 R_{2}$
(b) $R_{1}=3 R_{2}$
(c) $2 R_{1}=R_{2}$
(d) $3 R_{1}=R_{2}$

Key Concepts

Function Symmetry

This concept refers to a function having the property that its values mirror each other about a certain point or line. In the given problem, the condition f(x) = f(1-x) illustrates symmetry about the line x = 1/2. Such symmetry can greatly simplify problems in integration, especially when evaluating areas and moments, because it allows the splitting or rearranging of the integral domains based on the function's invariant behavior under reflection.

Definite Integration

Definite integration is the process of calculating the accumulation of quantities, such as area under a curve, over a specified interval. In the problem, two definite integrals are involved: one that computes the total area under the curve (R2) and another that computes the weighted average (or moment) of the area with respect to x (R1). Understanding definite integrals is central to solving problems involving cumulative quantities.

Centroid and First Moment (Moment of Area)

The centroid of a region is the point at which the entire area could be considered to be concentrated, and it is found by dividing the first moment of the area by the total area. In this context, R1 represents the first moment of the area and R2 represents the total area. Recognizing that the symmetry of the function forces the centroid to lie on a specific line (here, x = 1/2) is crucial in relating the two integrals and solving for the relationship between R1 and R2.

Change of Variables (Substitution Technique)

The substitution technique in integration involves replacing the variable of integration with another variable to simplify the integral or reveal underlying symmetries. In problems where a function satisfies a symmetry condition like f(x) = f(1-x), performing a substitution (for example, u = 1-x) can underscore the symmetry and lead to easier calculation or cancellation of terms in the integral.

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