In the adjacent figure the graph of two function `y=f(x)` and `y=sin x` are given y=sin x intersects, y=f(x) at A(a,f(a)),`B(pi,0) and C(2pi,0)`.
`A_(i)(i=1,2,3)` is the area bounded by the curves `y=f(x) and y=sin x` between x=0 and x=a,i=1 between x=a and `x=pi, i=2` between `x=pi` and `x=2pi,i=3`. If `A_(1)=1-sin a+(a-1)` cos a, determine the function f(x). Hence, determine a and `A_(1)`. Also, calculate `A_(2) and A_(3)`.
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Answer

` f(x) = x sin x, a = 1 ; A_(1) = 1 – sin 1 ; A_(2) = pi – 1 – sin 1 ; A_(3) = (3pi - 2) `sq. units