The base of a solid in the first quadrant of the xy plane is a right triangle bounded by the coordinate axes and the line x+y =4. Cross sections of the solid perpendicular to the base are squares. What is the volume, in cubic units , of the solid?

Question

A) 8
B) 32/3
C) 64/3
D) 128/3

Answer 1

perpendicular to a the base is not good enough.
Let's say the squares are perpendicular to the x-axis. Then the length of each side of the square is just the y-coordinate: 4-x
So, adding up the volumes of all those thin plates of thickness dx, we have
v = ∫[0,4] (4-x)^2 dx = 64/3

the same applies if the squares are perpendicular to the y-axis, due to the symmetry of the region.