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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)`

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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CENGAGE-DEFINITE INTEGRATION -Exercise (Multiple)
  1. If f(x) is integrable over [1,2] then int(1)^(2)f(x)dx is equal to

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  2. If L=lim(nto oo) (n^(3)(e^(1//n)+e^(2//n)+………+e))/((n+1)^(m)(1^(m)+4^(...

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  3. Let p=1+1/(sqrt(2))+1/(sqrt(3))+….+1/(sqrt(120)) and q=1/(sqrt(2))+1/(...

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  4. Let Sn=sum(k=0)^n n/(n^2+k n+k^2) and Tn=sum(k=0)^(n-1)n/(n^2+k n+k...

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  5. T h ev a l u eofint0^1(2x^2+3x+3)/((x+1)(x^2+2x+2))dxi s pi/4+2log2-...

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  6. Let f(x)=int(1)^(x)(3^(t))/(1+t^(2))dt, where xgt0, Then

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  7. If inta^b|sinx|dx=8 and int0^(a+b)|cosx| dx=9 , then find the val...

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  8. Iff(x)=int0^x2|t|dt ,t h e n g(x)=x|x| g(x) is monotonic g(x) is di...

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  9. IfAn=int0^(pi/2)(sin(2n-1)x)/(sinx)dx ,bn=int0^(pi/2)((sinn x)/(sinx))...

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  10. T h ev a l u eofint0^oo(dx)/(1+x^4)i s s a m ea st h a tofint0^oo(x^...

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  11. The value of int0^1e^x^(2-x)dx is <1 (b) >1 (c) > e^(-1/4) (d) <e^(...

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  12. If int(a)^(b)(f(x))/(f(x)+f(a+b-x))dx=10, then

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  13. The values of a for which the integral int0^2|x-a|dxgeq1 is satisfied ...

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  14. If f(x)=int(0)^(x)|t-1|dt, where 0lexle2, then

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  15. If f(2-x)=f(2+x) and f(4x)=f(4+x) for all x and f(x) is a function for...

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  16. Off(x)=int0^x("cos"(sint)+"cos"(cost)dt ,t h e nf(x+pi)i s f(x)+f(pi)...

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  17. If I(n)=int(0)^(pi//4) tan^(n)x dx, (ngt1 is an integer ), then

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  18. IfIn=int0^1(dx)/((1+x^2)^n),w h e r en in N , which of the following...

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  19. L e tf:[1,oo)vecRa n df(x)=int1^x(e^t)/t dt-e^xdotT h e n f(x) is an ...

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  20. Iff(x)=inta^x[f(x)]^(-1)dxa n dinta^1[f(x)]^(-1)dx=sqrt(2),t h e n f(...

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