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If A(n) is the area bounded by y=x and y...

If `A_(n)` is the area bounded by y=x and `y=x^(n), n in N,` then `A_(2).A_(3)…A_(n)=`

A

`(1)/(n(n+1))`

B

`(1)/(2^(n)n(n+1))`

C

`(1)/(2^(n-1)n(n+1))`

D

`(1)/(2^(n-2)n(n+1))`

Text Solution

Verified by Experts

The correct Answer is:
D
`y=x" intersect "y=x^(n)" at "(0,0) and (1,0)" for all " n in N`

`"Area "A_(n)=overset(1)underset(0)int(x-x^(n))dx`
`=[(x^(2))/(2)-(x^(n+1))/(n+1)]_(0)^(1)`
`=(1)/(2)-(1)/(n+1)`
`=(n-1)/(2(n+1))`
`"Thus, "A_(2)cdotA_(3)cdotA_(4)....A_(n)=(1)/(2^(n-1))((1)/(3)cdot(2)/(4)cdot(3)/(5)...(n-1)/(n+1))`
`=(1)/(2^(n-2).n(n+1))`
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CENGAGE-AREA-Exercise (Single)
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  2. Area enclosed between the curves |y|=1-x^2a n dx^2+y^2=1 is (3pi-8)/3 ...

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  3. If A(n) is the area bounded by y=x and y=x^(n), n in N, then A(2).A(3)...

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  4. The area of the region is 1st quadrant bounded by the y-axis, y=(x)/(4...

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  5. The area of the closed figure bounded by y=(x^2)/2-2x+2 and the tangen...

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  6. The area of the region bounded by x^(2)+y^(2)-2x-3=0 and y=|x|+1 is

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  7. The area enclosed by the curve y=sqrt(4-x^2),ygeqsqrt(2)sin((xpi)/(2sq...

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  8. The area bounded by the curve y^(2)=1-x and the lines y=(|x|)/(x),x=-...

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  9. The area bounded by the curves y=(log)e xa n dy=((log)e x)^2 is e-2s q...

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  10. The area bounded by y = 3-|3-x| and y=6/(|x+1|) is

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  11. Find the area enclosed between the curves: y = loge (x + e) , x = loge...

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  12. Find the area enclosed the curve y=sin x and the X-axis between x=0 an...

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  13. The area bounded by y=x^(2),y=[x+1], 0 le x le 2 and the y-axis is whe...

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  14. The area of the region bounded by the parabola (y-2)^(2) = x- 1, the t...

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  15. The area bounded by the curves y=x e^x ,y=x e^(-x) and the line x=1 is...

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  16. The area of the region whose boundaries are defined by the curves y=2c...

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  17. Area bounded by y=sec^-1x,y=cot^-1x and line x=1 is given by

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  18. The area bounded by the curve y=3/|x| and y+|2-x|=2 is

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  19. The area enclosed by y=x^(2)+ cos x" and its normal at "x=(pi)/(2) in ...

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  20. "Given "f(x)=int(0)^(x)e^(t)(log(e)sec t- sec^(2)t)dt, g(x)=-2e^(x) ta...

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