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Sketch the region bounded by the curves ...

Sketch the region bounded by the curves `y=x^(2) and y=(2)/(1+x^(2))`. Find the area.

Text Solution

Verified by Experts

The correct Answer is:
`(pi-(2)/(3))` sq unit
The curve `y=x^(2) ` is a parabola. It is symmetric about Y-axis and has its vertex at (0, 0) and the curve `y=(2)/(1+x^(2))` is a bell shaped curve . X-axis is its asymptote and it is symmetric about Y-axis and its vertex is (0, 2).

Since, `y=x^(2) " " ` ... (i)
and `y= (2)/ (1+x^(2) ) " " ` ... (ii)
` rArr y= (2)/ ( 1+y) `
`rArr y^(2) +y-2=0`
` rArr (y-1)(y+2)=0 rArr y= -2, 1`
But ` y ge 0, so y=1 rArr x= pm 1`
Therefore , coordinates of C are (-1, 1) and coordinates of B are (1, 1) .
` therefore " Required area " OBACO = 2 xx " Area of curve " OBAO`
`=2 [int_(0)^(1) (2)/(1+x^(2))dx - int_(0)^(1)x^(2)dx]`
`2[[2tan^(-1)x]_(0)^(1) - [(x^(3))/(3)]_(0)^(1)]=2[(2pi)/(4)-(1)/(3)]=(pi-(2)/(3))` sq unit
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