`x(t) = ( cos(2t), -sin(2t)) for time 0 <= t <= pi/4`

traverses an anti-clockwise quarter-circle around the origin from (1,0)
to (0,-1).

Question 2: The path

x`(t) = (1-t, t) for time 0 <= t <= 1`

will traverse a line segment from (1,0) to (0,1). Other parameterizations are possible, e.g.

`x(t) = (1-2t,2t) for time 0 <= t <= 1/2.`

Question 3: Each term has a magnitude which is

if

Question 4: Integrating

`(partial u(x,y))/(partial x) = x y^2`

we get

`u(x,y) = x^2 y^2/2 + f(y)`

where `f(y)` is some undetermined function of `y`.
Putting this back into the other equation we get

`x^2 y = x^2 y + f'(y)`

so f is actually a constant, i.e.

`u(x,y) = x^2 y^2/2 + C.`

Substituting `x=0,y=0` we get `C=1`, so `u(1,1) = 3/2`.

Question 5: Since

`ln|-2| - ln|-4| = ln 2 - ln 4 = ln (1/2) = - ln 2.`

Question 6: Since

`1/(100+x^3) <= 1/99.`

Since the integral of 1/99 from -1 to 1 is clearly equal to 2/99, we are done.