The x term represents the distance between an arbitrary x and the axis of rotation.

If the solid is rotated about the yaxis the x term is just x.

In your example the distance between x and the axis of rotation x = 1 is

1-x. so replace x with 1-x--- you obtain (1-x)*(sqrt(x)-x^3) for the integrand.

A more general form of the shell method is 2pint{D(x)*(f-g)dx}

Where D(x) is as mentioned the distance between an arbitrary x and the axis of rotation