Question

Let $f:[-2,3]\to [0,\infty )$ be a continuous function such that $f(1-x)-f\left(x\right)$ for all $xϵ[-2,3]$.

If $R_1$ is the numerical value of the area of the region bounded by $y=f\left(x\right),x=-2,x=3$ and the axis of x and $R_2={\int }_2^{3}xf\left(x\right)dx$, then:-

Answer 1

$R_1={\int }_{-2}^{3}f\left(x\right)dx$

$R_2={\int }_{-2}^{3}xf\left(x\right)dx$

$={\int }_{-2}^3\frac{xf\left(x\right)+(1-x)f(1-x)}{2}dx$

$={\int }_{-2}^3\frac{xf\left(x\right)+(1-x)f\left(x\right)}{2}dx$

$=\frac{1}{3}{\int }_{-2}^{3}f\left(x\right)dx=\frac{R_1}{2}$