Question

Let f : [- 4, 5] $\to$ [0, $\infty$) be a continuous function such that f(1 - x) = f(x) $\forall$ for all x $ϵ$ [- 4, 5]. If $R_1$ is the numerical value of area of the region bounded by y = f(x), x = - 4 and x = 5 and x-axis, $R_2={\int }_{-4}^{5}xf\left(x\right)dx$, then

• A. $R_1=R_2$
• B. $R_1=2R_2$
• C. $2R_1=3R_2$
• D. $3R_1=2R_2$

Answer 1

The correct option is B $R_1=2R_2$

$R_1={\int }_{-4}^{5}dx$ = area of region bounded by x = -4, x = 5, y = f(x) and x-axis
Given $R_2={\int }_{-4}^{5}xf\left(x\right)dx={\int }_{-4}^5(1-x)f(1-x)dx(\because f(1-x)=f(x\left)\forall xϵ\right[-4,5\left]\right)$
$\therefore R_2={\int }_{-4}^{5}f(1-x)f\left(x\right)={\int }_{-4}^{5}f\left(x\right)dx-{\int }_{-4}^{5}xf\left(x\right)dx=R_1-R_2$
$\Rightarrow 2R_2=R_1$