Question

Let `f:[1,2] to [0,oo)` be a continuous function such that `f(x)=f(1-x)` for all `x in [-1,2]. ` Let `R_(1)=int_(-1)^(2) xf(x) dx,` and `R_(2)` be the area of the region bounded by `y=f(x),x=-1,x=2` and the x-axis . Then,
A. `R_(1)=2R_(2)`
B. `R_(1)=3R_(2)`
C. `2R_(1)=3R_(2)`
D. `3R_(1)=R_(2)`

Answer 1

Correct Answer - c
We have ,
`R_(1)=underset(-1)overset(2)int xf(f) dx`
Using :` underset(a)overset(b)intf(a+b-x)dx`, we obtain
`R_(1)=underset(-1)overset(2)int (1-x)f(1-x)dx`
`implies R_(1) = underset(-1)overset(2)int(1-x)f(x) dx " "[because f(1-x)=f(x) "(given")"]"`
`implies R_(1)=underset(-1)overset(2)int f(x) dx-R_(1)implies 2R_(1)=underset(-1)overset(2)f(x) dx`
It is given that `R_(2)=underset(-1)overset(2)f(x) dx`