Concept preview

What you will be able to do

1. 01Extend $V=\displaystyle\int_a^b A(x)\,dx$ to solids whose cross sections are triangles or semicircles, still reading the side length $s(x)$ off the base as top minus bottom.
2. 02Apply the correct area formula for each face: equilateral triangle $A=\tfrac{\sqrt3}{4}s^2$, isosceles-right triangle, and semicircle $A=\tfrac{\pi}{8}s^2$ when the diameter equals $s$ — keeping $\pi$ and $\sqrt3$ symbolic.
3. 03Set up and evaluate the volume integral to a clean symbolic answer, and recognise the classic slip of using the WRONG area formula (for example treating the base side as a radius instead of a diameter).

The engine is exactly the one from 8.7. A solid with known cross sections has volume $V=\int_a^b A(x)\,dx,$ where $A(x)$ is the area of the slice at position $x$. When the slabs are perpendicular to the $x$-axis, the side that sits on the base is still $s(x)=f(x)-g(x)$, the top curve minus the bottom curve. The only change from the last lesson is the function that turns that side into an area. For squares it was $A=s^2$; now we need the area formulas for triangles and semicircles built on a base of length $s$.