Volumes by Slicing, Disks, and Cylindrical Shells

The same slice-approximate-integrate logic that gives the area between two curves extends to three dimensions: cut a solid into thin slabs, approximate each slab as a cylinder (volume = base area × thickness), sum, and take the limit. The result is $V = \int_a^b A(x)\,dx$ where $A(x)$ is the cross-sectional area. When the solid is generated by rotating a curve around an axis, the cross-section is a disk or washer, giving the disk/washer method. The cylindrical shell method integrates thin shells instead of disks — it is often easier when rotating around the $y$-axis. While volumes of revolution are not themselves central to quantitative finance, the skill of setting up a triple of limits from a geometric description and converting a 3D problem to a 1D integral is directly transferable.

Learning objectives. By the end of this note you should be able to: