Let `f:[-1,2]vec[0,oo)` be a continuous function such that `f(x)=f(1-x)fora l lx in [-1,2]dot` Let `R_1=int_(-1)^2xf(x)dx ,` and `R_2` be the area of the region bounded by `y=f(x),x=-1,x=2,` and the `x-a xi s` . Then `R_1=2R_2` (b) `R_1=3R_2` `2R_1` (d) `3R_1=R_2`

Options

A. `R_(1)=2R_(2)`
B. `R_(1)=3R_(2)`
C. `2R_(1)=R_(2)`
D. `3R_(1)=R_(2)`

Answer

C

Explanation

`R_(1)=int_(-1)^(2)xf(x)dx=int_(-1)^(2)(2-1-x)f(2-1-x)dx`
`=int_(-1)^(2)(1-x)f(1-x)dx`
`=int_(-1)^(2)(1-x)f(x)dx`
Hence `2R_(1)=int_(-1)^(2)f(x)dx=R_(2)`.