Good night. I have a question on what parameter to use when evaluating line integrals.

I was practicing some questions, asking to find the intergral of F dr over the region C where it was defined as a piecewise region C1 U C2.

C1 was the straight line path from points P1 (1,1), to P2 (1,3)

C2 was y=x^2 2, moving from P2 (1,3), to P3 (2,6)

given the Function F(x,y)= (x y)i-(1)j

When I first attempted it I tried using the equation Parameter=(Point) t<vector>

So I found the Vector

P1P2= (1,3)-(1,1)=<0,2>

Then used Point 1 on the line and that vector in the parameter equation resulting in

r(t)=p tv

r(t)=(1,1) t<0,2>

r(t)=<1,1 2t,>

Giving x=1 , and y=1 2t to plug into F(x,y) giving

F(t)=((1) (1 2t))i -(1)j

F(t)=<2 2t,-1>

and F'(t)=<0,2>

So for the C1 part plugging it into the equation I got

Integral (from 1 to 3)<2 2t,-1><0,2> dt

which i got as

int C1 (F)dr= -4
However looking back at the solution given, I have no clue where they got the perameters as

x(t)=1

y(t)=t

and they got the answer as -2

https://preview.redd.it/ku8s0s3hvzbd1.png?width=396&format=png&auto=webp&s=726a77a113fe434084c8851404079f403e28afef

Could anyone explain my error, or How they got the answer as -2? Did I pick the wrong perameter? What should I have done here

I appreciate the assistance anyone can give here.

*EDIT* This is my revised working. Im not sure what mistake i made to get -1/6 as opposed to 5/6 like the sample solution above