Find the work done by f= (x^2+y)i + (y^2+x)j +(ze^z)k over the following path from (4,0,0) to (4,0,4)

a. the line segment x=4 y=0 z=[0; 4]

find a scalar potential function f for f, such that

f=∇f.

the work done by f over the line segment is?

b. the helix r(t)= (4cost)i +(4sint)j +(2t/pi)k t=[0; 2pi]

find df/dt for f?

the work done by f over the helix?

c. the x axis from (4,0,0) to (0,0,0) followed by the line z=x, y=0 from (0,0,0) to (4,0,4)

what is the integral to comput the work done by f along the x-axis from followed by the line z=x?

what is the work done by f over the 2 curves

We want to find such that . This means

Integrating both sides of the latter equation with respect to tells us

and differentiating with respect to gives

Integrating both sides with respect to gives

Then

and differentiating both sides with respect to gives

So the scalar potential function is

By the fundamental theorem of calculus, the work done by along any path depends only on the endpoints of that path. In particular, the work done over the line segment (call it ) in part (a) is

and does the same amount of work over both of the other paths.

In part (b), I don't know what is meant by "df/dt for F"...

In part (c), you're asked to find the work over the 2 parts (call them and ) of the given path. Using the fundamental theorem makes this trivial: