How to find right parameter and calculate the work done

Question

Hi I am having trouble calculating the work done in moving a particle from $(-1,2,5)$ to $(1,0,1)$ where $F=yi+xj+zk$ on the curve C, where the curve C is the intersection of $z=x^2+y^2$ and the plane $x+y=1$

I know I have to parametrize to get the right r(t) and then I can use that to calculate the work done. But I don't know how to parametrize it. Would it just along z=1?

Please help. Thanks

Answer 1

Set $x=t$. $x + y = 1 \implies y = 1-t$. $z = x^2 + y^2 \implies z = t^2 + (1-t)^2$. Therefore one parameterization is

$${r}(t) = <t, 1-t, t^2 + (1-t)^2>; -1 \leq t \leq 1 $$

You can calculate work done using this parameterization as follows:

$$W = \int_C \mathbf{F} \cdot \mathbf{dr} = \int_{t_1}^{t_2} \mathbf{F}(r(t)) \cdot r'(t) dt$$