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Sketch the curves and identify the regio...

Sketch the curves and identify the region bounded by the curves `x=1/2,x=2,y=logxa ny=2^xdot` Find the area of this region.

Text Solution

Verified by Experts

The correct Answer is:
`((4-sqrt(2))/log 2-(5)/(2) log 2+(3)/(2))`sq. units
The given curves are
`x=(1)/(2)`
`x=2`
`y=log_(e)x`
`y=2^(x)`

Required area = Area of ABCDA
`=int_(1//2)^(2)(2^(x)-log x)dx`
`[(2^(x))/(log 2)-(x log x -x)]_(1//2)^(2)`
`=((4)/(log 2)-2 log 2+2)-(-(sqrt(2))/(log 2)-(1)/(2)log""(1)/(2)+(1)/(2))`
`=((4-sqrt(2))/(log 2)-(5)/(2)log 2 +(3)/(5))` sq. units.
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