Volumes with Cross Sections: Squares and Rectangles

Picture a flat region in the xy-plane as the footprint of a solid. Instead of the solid having uniform thickness, you build it by standing a shape, a square or a rectangle, straight up on every vertical sliver of that region. The side length of each shape comes from the height of the curve at that x-value. Slide along the x-axis and the shape changes size as the curve rises and falls, so the solid bulges and narrows to match.

To get the volume, write the area of one cross section as a function of x, call it A(x), then integrate A(x) dx across the interval that defines the region. This works because a thin slab of thickness dx has volume approximately A(x) dx, and summing infinitely many of these slabs is exactly what the integral does. For squares standing on y = f(x), the side length is f(x), so A(x) = [f(x)]^2. For rectangles with a fixed height-to-width ratio k, the width is still f(x) but the height is k times that, so A(x) = k[f(x)]^2.

This topic is the simplest case of a bigger family: volumes by cross sections, which later includes semicircles, equilateral triangles, and the disk and washer methods built on circular cross sections. The setup is identical every time, find A(x), integrate it, only the shape formula for the area changes. Mastering squares and rectangles here makes every other cross-section shape just a matter of swapping one formula.