Identify the radius and height of the shell, and compute the volume for the region bounded by y=x^2 and y=0

I think I'm being tricked in my calculus 2 class..

This is #6 in my book:

Sketch the region, draw in a typical shell, identify the radius and height of the shell, and compute the volume for the region bounded by y=x^2 and y=0, -1
And here's #1 in my book:


Sketch the region, draw in a typical shell, identify the radius and height of the shell, and compute the volume for the region bounded by y=x^2 and the x-axis, -1
The difference is instead of x-axis it's y=0. Do you do both problems the same way?

Yes, you do both problems the same way. I guess the author wants to make sure you know that y = 0 and the x-axis are the same?

Image for #6 (and #1): http://i.imgur.com/3ZFJZ.png

As you can see, each shell has a radius of (2 - x) and a height of (y) = (x²).

The volume via the shell method is described by:
V = 2π ∫₋₁¹ (2 - x) x² dx,

or, taking advantage of symmetry:
V = 2 ∙ 2π ∫₀¹ (2 - x) x² dx
V = 4π ∫₀¹ (2x² - x³) dx

V = 4π [2/3 x³ - 1/4 x⁴]₀¹
V = 4π [(2/3 - 1/4) - (0 - 0)]
V = 4π (5/12)
V = 5π/3

Edit: left out the 2 on the first integral. Sorry.