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The area of the region for which 0<y<<3-...

The area of the region for which `0>0` is

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`0ltylt3-2x-x^(2),xgt0`
`"Consider parablola "y=3-2x-x^(2)`
`" or "(x+1)^(2)=-(y-2)`
The parabola having vertex `(-1,2)` and concave downward.
For `x,ygt0` the points lie inside parabola in the first quadrant as shown in the following figure.

`therefore" Required area"=overset(1)underset(0)int(3-2x-x^(2))dx`
`=[3x-x^(2)-(x^(3))/(3)]_(0)^(1)=(5)/(3)" sq. units"`
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