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If S(0),S(1),S(2),… are areas bounded by...

If `S_(0),S_(1),S_(2),…` are areas bounded by the x-axis and half-wave of the curve `y=sin pi sqrt(x)," then prove that "S_(0),S_(1),S_(2),…` are in A.P…

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`y= sin pi sqrt(x)` meets x-axis when `pisqrt(x)=npi or x=n^(2), n in N.` Therefore, area of half-wave between `x=n^(2) and x=(n+1)^(2)` is
`S_(n)=|overset((n+1)^(2))underset(n^(2))int sin pi sqrt(x)dx |`
`"Putting "pisqrt(x)=y and pi^(2) dx =2y dy,`we get
`therefore" "S_(n)=|(2)/(pi^(2))overset((n+1)pi)underset(npi)inty sin y dy |`
`=|(2)/(pi^(2))[-y cos y + sin y ]_(npi)^((n+1)pi)|`
`=|(2)/(pi^(2))[-(n+1)pi cos (n+1) pi +npi cos n pi ]|`
`=(2(2n+1))/(pi), n in N`
`"Hence, "S_(0),S_(1),S_(2),...` are in A.P..
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