Volumes with Cross Sections: Triangles and Semicircles

This is the same slicing idea as volumes by disks, but the slices are no longer circles from revolving a region. Instead you're told the shape of each cross section directly, and that shape sits on top of a base region like the one between two curves. The height of the region at each x becomes some length in the slice: a diameter for a semicircle, a side length for a triangle. Different shapes built on the exact same base give different volumes, because a semicircle and a square with the same width don't have the same area. The base only tells you the width of each slice; you still have to supply the geometry.

The mechanism is always the same three steps. Find the length across the base at position x — call it w(x), usually a difference of two functions like √x minus 0, or x minus x^2. Use that length to write the area of one slice, A(x), using the right formula for the shape. Then integrate A(x) dx over the interval to add up all the infinitely thin slices. For a semicircle with diameter w(x), the radius is w(x)/2, so A(x) = (1/2) π (w(x)/2)^2. For an equilateral triangle with side w(x), A(x) = (√3/4) w(x)^2.

This topic sits right next to square and rectangle cross sections, and the setup barely changes between them — only the area formula does. It all builds on the same accumulation principle as every integral in this unit: break a solid into thin pieces, write down one piece's measure, and integrate. Getting comfortable with swapping in different area formulas here makes volumes by disks and washers, later generalizations of this same slicing idea, feel like old news.