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Find the area bounded by the curve xy^(2...

Find the area bounded by the curve `xy^(2)=4(2-x)` and y-axis.

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The correct Answer is:
`4pi` sq. units
We have curve `xy^(2)=4(2-x)`
`therefore" "x=(8)/(y^(2)+4)`
The given curve is symmetrical about x-axis and meets it at (2,0).
`"Also, when "yrarrpmoo,xrarr0.`
So, y-axis is asymptote to the curve.
The graph of the function is as shown in the following figure.

The line x=0, i.e., y-axis is asymptote.
Area of the shaded region `=2int_(0)^(oo)(8)/(y^(2)+4)dx`
`=16[(1)/(2)tan^(-1)""(y)/(2)]_(0)^(oo)=8xx(pi)/(2)`
`=4pi` sq. units
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