Path Integral of F = (x, y2, 2z) from a to b - Calculating Line Integrals

[Prologue]

Homework Statement

For F = (x, y², 2z), evaluate the path integrals along the line of a to b:

$$\vec{a}=(0,0,0), \vec{b}=(1,1,1), \int^{b}_{a} \vec{F} \times d\vec{r}$$

$$\int^{b}_{a} \vec{F} ds$$

Homework Equations

No idea.

The Attempt at a Solution

I don't have a clue what these integrals even evaluate to. The first one should be a vector, and I have no idea what that even means, an integral that isn't a scalar. The second one is the same problem.

 

[Dick]

Write down the path r(s)=(s,s,s) from (0,0,0) to (1,1,1) with s going from 0 to 1. dr is (dr/ds)*ds which is (1,1,1)*ds. The integral of Fds is F(r(s))*ds. To integrate the vector ds just find the vector whose components are the integral of each component of the vector. I.e. (integral sds, integral s^2ds, integral 2sds). To find Fxdr cross the vector F(r(s)) with (1,1,1) and integrate that vector ds.