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If |underset(a)overset(b) intf(x)dx|=und...

If `|underset(a)overset(b) intf(x)dx|=underset(a)overset(b) int|f(x)|dx,a ltb,"then " f(x)=0` has

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`" Let " I= int_(1)^(b) x f(x) dx" "...(1)`
`rArr I= int_(a)^(b) (a+b-x)f(a+b-x)dx`
`=int_(a)^(b) (a+b-x)f(x) dx" "....(2)`
`[ :' f(a+b-x)=f(x)" given "]`
Adding equations (1) and (2)
`rArr 2I = int_(a)^(b) (a+b) f(x) dx`
` I=(a+b)/(2)int_(a)^(b) f(x) dx`
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