Find the area bounded by the parabola `y=x^2+1` and the straight line `x+y=3.`

Explanation

The two curves meet at points where `3-x=x^(2)+1`,
`i.e.," "x^(2)+x-2=0`
`"or "(x+2)(x-1)=0 or x=-2,1`
`therefore" Required area "=int_(-2)^(1)[(3-x)-(x^(2)+1)]dx`
`=int_(-2)^(1)(2-x-x^(2))dx`
`=[2x-(x^(2))/(2)-(x^(3))/(3)]_(-2)^(1)`
`=(2-(1)/(2)-(1)/(3))-(-4-(4)/(2)+(8)/(3))`
`=(9)/(2)` sq. units