My first question is when I am given two curves and are asked to find the area between them using the washer method, how do I determine the outer vs inner function? For example:
find the area between the curves $y=x^3, y=x, x \geq 0$ ROTATED ABOUT THE X-AXIS
Obviously I set them equal and get my points which are 1, 0. I guessed that the outer function was $y=x$ because at the point $x=.5$ $y=x \gt y=x^3$ Is this the correct reasoning?
Secondly I am one number off my answer and I am not sure why:
$$\int \pi(x)^2-\pi(x^3)^2 dx = \pi \int x^2-x^6 dx$$
which of course gives me $\pi[\frac{1}{3}-\frac{1}{7} \Big\vert_0^1]\pi$
I get an answer of $\frac{4\pi}{21}$
The answer in the book is $\frac{\pi}{4}$
Thirdly how on earth do I decide whether to use the cylindrical shell method or the disk method?
Your volume integral using the washer method is correct.
For most rotation volume problem, including this one, one could use either method and get the same result. For the problem in hand, the washer method is
$$\int_0^1\pi(y_2^2-y_1^2)dx=\pi\int_0^1(x^2-x^6)dx=\frac{4\pi}{21}$$
and the cylinder method is
$$\int_0^1 2\pi y(x_2-x_1)dy = 2\pi \int_0^1 y(y^{1/3}-y)dy=\frac{4\pi}{21}$$