Question 1: The path

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 2|2x-1| times as large than the previous one.  If 2|2x-1| < 1, then the series converges by the ratio test;
if 2|2x-1| > 1, then the series diverges by the ratio test.  When 2|2x-1| = 1, then each term has the same magnitude, and one does divergence by the zero test.  Thus one only has convergence when 2|2x-1| < 1, i.e. when 0 < x < 1.
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|x| is an anti-derivative of 1/x for all x other than 0, we have from the Fundamental theorem of Calculus that the integral is

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

Question 6: Since x ranges between -1 and 1, x^3 ranges between -1 and 1 and so 100 + x^3 ranges between 99 and 101.  Thus

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.