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

If f(x) is integrable over [1,2] then `underset(1)overset(2)(f)f(x)dx` is equal to

A

`underset(nto oo)Lim(1)/(n)underset(r=1)overset(n)sumf((r)/(n))`

B

`underset(nto oo)Lim(1)/(2n)underset(r=n+1)overset(n)sumf((r)/(n))`

C

`underset(nto oo)Lim(1)/(n)underset(r=1)overset(n)sumf((r+n)/(n))`

D

`underset(nto oo)Lim(1)/(n)underset(r=1)overset(2n)sumf((r)/(n))`

Text Solution

Verified by Experts

The correct Answer is:
A
`underset(nto oo)Lim(1)/(n)underset(r=n+1)overset(2n)sumf((r)/(n))underset(1)overset(2)(f)(x)dx`
`underset(nto oo)Lim(1)/(n)underset(r=1)overset(2n)sum"f"((r+n)/(n))=underset(0)overset(1)(f)f(1+x)dxunderset(1)overset(2)(f)f(t)dt=underset(1)overset(2)(f)f(x)dx`
`underset(nto oo)Lim(1)/(n)underset(r=1)overset(n)sumf((r)/(n))=underset(0)overset(1)(f)f(x)dx`
`underset(nto oo)Lim(1)/(n)underset(r=1)overset(n)sumf((r)/(n))=underset(0)overset(2)(f)f(x)dx`
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