If `f(x) = max {sin x, cos x,1/2},` then the area of the region bounde – InterviewSolution

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1.	If `f(x) = max {sin x, cos x,1/2},` then the area of the region bounded by the curves `y =f(x),` x-axis, Y-axis and `x=(5pi)/3` isA. `sqrt2-sqrt3+(5pi)/(12)`B. `sqrt2+(sqrt3)/(2)+(5pi)/(12)`C. `sqrt2+sqrt3+(5pi)/(12)`D. none of these
Answer» Correct Answer - b `f(x)=max{sinx,cosx,1/2} ={{:(cos",",0lexle(pi)/(4)),(sinx",",(pi)/(4)lexle(5pi)/(6)),((1)/(2)",",(5pi)/(6)lexle(5pi)/(3)):}` Let A be the required area. Then, `A=underset(0)overset(5pi//3)(int)f(x)dx` `impliesA=underset(0)overset(pi//4)(int)cosx dx+underset(pi//4)overset(5pi//6)(int)sinx dx+underset(5pi//6)overset(5pi//3)(int)1/2dx` `impliesA=[sinx]_(0)^(pi//4)-[cosx]_(pi//4)^(5pi//6)+1/6((5pi)/(3)-(5pi)/(6))` `impliesA=(1)/(sqrt2)-(-(sqrt3)/(2)-(1)/(sqrt2))+(5pi)/(12)=(5pi)/(12)+(sqrt3)/(2)+sqrt2`

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