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Find the area of the region bounded by t...

Find the area of the region bounded by the curve `C : y=tanx ,t a nge n td r a w ntoC` at `x=pi/4,` and the x-axis.

Text Solution

Verified by Experts

The correct Answer is:
`[(log sqrt(2) -(1)/(4))" sq units "]`
Given, `y= tanx rArr (dy)/(dx)=sec^(2)x `
`therefore ((dy)/(dx))_(x=(pi)/(4))=2`
Hence, equation of tangent at `A((pi)/(4),1)` is
`(y-1)/(x-pi//4)=2 rArr y-1=2x-(pi)/(2)`

` rArr (2x-y) = ((pi)/(2)-1)`
` therefore ` Required area is OABO
`=int_(0)^(pi//4)(tanx)dx -"area of "Delta ALB`
`=[log|sec x|]_(0)^(pi//4)-(1)/(2).BL.AL`
`=log sqrt(2)-(1)/(2)((pi)/(4)-(pi-2)/(4))*1`
`=(log sqrt(2) -(1)/(4))` sq unit
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