Question

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.

Answer 1

I assume you want the volume of the shape with the given cross sections. In the
`x,y`-plane, the base is set

`\left\{(x,y) ~:~ 0\le x\le 2 \text{ and } 0 \le y \le 2-x\right\}`

Each cross section is a square plate with side length
`2-x` (the vertical distance between the
`x`-axis and the line
`y=2-x`) and thickness
`\Delta x`, hence volume
`(2-x)^2\,\Delta x`.

As
`\Delta x\to0` and the number of cross sections grows to infinity, the total volume of the plates converges to volume of the solid
`S`, given by the definite integral

`\displaystyle \int_0^2 (2-x)^2 \, dx = \int_0^2 x^2 \, dx = \frac{2^3-0^3}3 = \boxed{\frac83}`

(where we make the substitution
`2-x\mapsto x`)