If f(x) is integrable over [1,2] then in...

If `f(x)` is integrable over `[1,2]` then `int_(1)^(2)f(x)dx` is equal to (a) `lim_(nto oo) 1/n sum_(r=1)^(n)f(r/n)` (b) `lim_(nto oo) 1/n sum_(r=n+1)^(2n) f(r/n)` (c) `lim_(nto oo) 1/n sum_(r=1)^(n)f((r+n)/n)` (d) `lim_(nto oo) 1/n sum_(r=1)^(2n)f(r/n)`

(a) `lim_(nto oo) 1/n sum_(r=1)^(n)f(r/n)`

`lim_(nto oo) 1/n sum_(r=n+1)^(2n) f(r/n)`

`lim_(nto oo) 1/n sum_(r=1)^(n)f((r+n)/n)`

`lim_(nto oo) 1/n sum_(r=1)^(2n)f(r/n)`

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The correct Answer is:

B, C

`lim_(nto oo) 1/n sum_(r=n+1)^(2n)f(r/n)=int_(1)^(2) f(x)dx`
`lim_(nto oo) 1/n sum_(r=1)^(n)f((r+n)/n)=int_(0)^(1)f(1+x)dx=int_(1)^(2)dx=int_(1)^(2)f(t)dt=int_(1)^(2)f(x)dx`
`lim_(nto oo) 1/n sum_(r=1)^(n)f(r/n)=int_(0)^(1)f(x)dx`
`lim_(n to oo) 1/n sum_(r=1)^(2n) f(r/n)=int_(0)^(2)f(x)dx`

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