Problem 1 on WeBWorK set 8.
Find the volume of the solid whose base is the region enclosed by y=(x^2)
 and y=1, and the cross sections perpendicular to the y-axis are squares.
 
 Note that it talked about the cross section perpendicular to the y-axis, not the x-axis.
 
 In this figure, the base is orange.  The cross section at y = (Subscript[y, i]) is a square with side length 2(Sqrt[(Subscript[y, i])]), so A((Subscript[y, i])) = 4 (Subscript[y, i]) and the volume of the slice between y = (Subscript[y, i]) and y = (Subscript[y, i]) + Δy is Δ(Subscript[V, i]) = 4 (Subscript[y, i]) Δy.  Adding the volume of all these slices, and taking the limit of Δy → 0 gives V = (∫_0^1)4ydy = 2

Graphics3D[ {(*base*) Polygon[Table[{x, (x^2), 0}, {x, -1,1,.1}]],(*face at y = 1 *)Polygon[{{1,1,0}, {1,1,2}, {-1,1,2}, {-1,1,0}}],
(* square slices at y = constant *), Thick,Table[Line[{{(Sqrt[y]),y,0},{(Sqrt[y]),y,2(Sqrt[y])}, {-(Sqrt[y]),y,2(Sqrt[y])}, {-(Sqrt[y]),y,0}, {(Sqrt[y]),y,0}}], {y, .01,1, .1} ],
(* edge on right *)
Line[Table[{(Sqrt[y]), y, 2(Sqrt[y])}, {y, 0, 1, .1}]],
(* edge on left *)
Line[Table[{-(Sqrt[y]), y, 2(Sqrt[y])}, {y, 0, 1, .01}]]}, PlotRange->{{-1,1}, {0,1}, {0,2}}, Axes -> True,AxesLabel ->{"x","y","z"} ]

For 3 points of extra credit, find A(x) for this solid and compute the volume by integrating V = (∫_(-1)^1)A(x)dx.
Submit a paper version of the computation to Prof. Swift by Friday, September 28 at the beginning of class.