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

If `f(x)` is integrable over `[1,2],` then `int_1^2f(x)dx` is equal to
(a) `("lim")_(ntooo)1/nsum_(r=1)^nf(r/n)` (b) `("lim")_(ntooo)1/nsum_(r=n+1)^(2n)f(r/n)` (c) `("lim")_(ntooo)1/nsum_(r=1)^nf((r+n)/n)` (d) `("lim")_(ntooo)1/nsum_(r=1)^(2n)f(r/n)`

A

(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)`

Text Solution

Verified by Experts

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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Knowledge Check

  • The value of lim_(n to oo)sum_(r=1)^(n)(1)/(n)e^((r)/(n)) is -

    A
    `1-e`
    B
    `e-1`
    C
    e
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  • The value of lim_(n to oo) sum_(r=1)^(n)(r^(2))/(r^(3)+n^(3)) is -

    A
    `(pi)/(4)`
    B
    `(pi)/(2)`
    C
    `(1)/(3)log_(e)2`
    D
    `(1)/(2)log_(e)2`

  • The value of lim_( n to oo) sum_(r =1)^(n) (r )/(n^(2))"sec"^(2)(r^(2))/(n^(2)) is equal to -

    A
    tan 1
    B
    `(1)/(2)tan1`
    C
    sec 1
    D
    `(1)/(2)sec1`

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