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For positive integers `K=1,2,3,……….n` Let `S_(k)` denotes the area of `DeltaOAB_(k)` (where `O` is the origin) such that `/_AOB_(k)=(Kpi)/(2n),OA=1` and `OB_(K)=K`. The value of the `lim_(nto oo)1/(n^(2)).sum_(K=1)^(n)S_(K)` is

A

`(2)/(pi^(2))`

B

`(4)//(pi^(2))`

C

`(8)/(pi^(2))`

D

`(1)/(2pi^(2))`

Text Solution

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The correct Answer is:
D
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VIKAS GUPTA (BLACK BOOK) ENGLISH-INDEFINITE AND DEFINITE INTEGRATION-EXERCISE (SUBJECTIVE TYPE PROBLEMS)
  1. For positive integers K=1,2,3,……….n Let S(k) denotes the area of Delta...

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  2. int (x+ ( cos^(-1)3x )^(2))/(sqrt(1-9x ^(2)))dx = (1)/(k (1)) ( sqrt(1...

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  3. If int (0)^(oo) (x ^(3))/((a ^(2)+ x ^(2)))dx = (1)/(ka ^(6)), then fi...

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  4. Let f (x) = x cos x, x in [(3pi)/(2), 2pi] and g (x) be its inverse. ...

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  5. If int (x ^(6) +x ^(4)+x^(2)) sqrt(2x ^(4) +3x ^(2)+6) dx = ((ax ^(6) ...

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  6. If the value of the definite integral int ^(-1) ^(1) cos ^(-1) ((1)/(s...

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  7. The value of int (tan x )/(tan ^(2) x + tan x+1)dx =x -(2)/(sqrtA) tan...

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  8. Let int (0)^(1) (4x ^(3) (1+(x ^(4)) ^(2010)))/((1+x^(4))^(2012))dx = ...

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  9. Let int ( 1) ^(sqrt5)(x ^(2x ^(2)+1) +ln "("x ^(2x ^(2x ^(2)+1))")")dx...

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  10. If int (dx )/(cos ^(3) x-sin ^(3))=A tan ^(-1) (f (x)) +bln |(sqrt2+f ...

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  11. Find the value of |a| for which the area of triangle included between ...

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  12. Let I = int (0) ^(pi) x ^(6) (pi-x) ^(8)dx, then (pi ^(15))/((""^(15) ...

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  13. If I = int (0) ^(100) (sqrtx)dx, then the value of (9I)/(155) is:

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  14. Let I(n) = int (0)^(pi) (sin (n + (1)/(2))x )/(sin ((x)/(2)))dx where ...

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  15. If M be the maximum value of 72 int (0) ^(y) sqrt(x ^(4) +(y-y^(2))^(2...

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  16. Find the number points where f (theta) = int (-1)^(1) (sin theta dx )/...

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  17. underset(nrarroo)lim[(1)/(sqrtn)+(1)/(sqrt(2n))+(1)/(sqrt(3n))+...+(1)...

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  18. The maximum value of int (-pi//2) ^(2pi//2) sin x. f (x) dx, subject t...

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  19. Given a function g, continous everywhere such that g (1)=5 and int (0)...

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  20. If f (n)= 1/pi int (0) ^(pi//2) (sin ^(2) (n theta) d theta)/(sin ^(2)...

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  21. Let f (2-x) =f (2+xand f (4-x )= f (4+x). Function f (x) satisfies int...

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