I am trying to evaluate a line integral over the boundary of the area in the x-y plane between the parabola $\displaystyle y=x^2 $ and the line $\displaystyle y=1 $, in an counterclockwise direction.

The integral is

Int((y^2-x)dx+(3x+y)dy) (still learning latex ...sorry)

Can someone please tell me: Do I have to split up the enclosed boundary into the straight line part and the parabola part, and do them separately? This is what I have done.

I have parametrised the line as $\displaystyle x=1-2t, y=1 from t =0 to1 $

and the parabola separately as $\displaystyle x=t, y=t^2 from t =-1 to 1 $.

I then evaluated the line integrals separately and got -2 and 4.4

Not sure if I need to now simply add them as they are or add absolute values (I'm thinking the first one)

Could someone kind out there please check my first value of -2. The problem I encountered is : If dy/dt=0 then dy=0dt and does that mean the second part of the integral (ie the bit with 3x+y) just becomes 0?

Any help appreciated. Thanks.