Question

The base of a certain solid is the triangle with vertices at (−12,6), (6,6), and the origin. Cross-sections perpendicular to the y-axis are squares.

Answer 1

The area of the solid is a set of vertical strips of width dx.
The other edges of the triangle are the lines y = -x/2 and y=x
So the volume of the solid is
v = ∫[-12,0] (6 + x/2)^2 dx + ∫[0,6] (6-x)^2 dx = 144 + 72 = 216

Answer 2

Oops. My bad. I took cross-sections perpendicular to the x-axis.
So see what you can do to fix my mistake, taking horizontal strips as the base of each square.