Let f:[-1,2] [0, ) be a continuous function such that f(x)=f(1-x) for all x [-1,2] . Let R₁= _ -1 ^2 x f(x) d x and R₂ be the area of the region bounded by y=f(x), x=-1, x=2 and the X -axis. Then,

Step-by-step solution

Here, R₁= _ -1 ^2 x f(x) d x Using, _a^b f(x) d x= _a^b f(a+b-x) d x aligned & R₁= _ -1 ^2(1-x) f(1-x) d x, & .R₁= _ -1 ^2(1-x) f(x) d x (1-x) ] aligned Given, R₂ is area bounded by aligned & f(x), x-1 and x=2 & R₂= _ -1 ^2 f(x) d x aligned Adding Eqs. (i) and (ii), we get 2 R₁= _ -1 ^2 f(x) d x From Eqs. (iii) and (iv), we get 2 R₁=R₂