Question

Consider the solid S described below. The base of S is the triangular region with vertices (0, 0), (2, 0), and (0, 2). Cross-sections perpendicular to the x-axis are squares. Find the volume V of this solid.

Answer 1

$S$ is the region in the $x,y$ plane bounded by lines $x=0$, $y=0$, and $x+y=2$ or $y=2-x$.

Each cross section, for a given value of $x$ in the interval [0, 2], has an area equal to $s^2$, where $s=(2-x)-0=2-x$ is the length of the square section's base.

Then the volume of the solid is

$\displaystyle\int_0^2(2-x)^2\,\mathrm dx=\boxed{\frac83}$