Course: Calculus II
Topic: Geometry
Subtopic: Volume by Slicing Methods


The previous lesson covered finding area of a bounded region in 2D. In this lesson we introduce using integrals to find the volume of certain 3D solids. 3D solids can take on a usual form such as a cylinder or cone. Known formulas such as `V=(pi r^2 h)/3` can be used to find their volume. But some 3D solids are unusual in their form, yet do have rotational symmetry about a central axis. The volume of such solids of revolution can be found using integration via what is called the slicing methods (method of disks and method of washers). The volume of solids that are not formed by revolving about an axis can be found by the cross section method where the volume of thin slabs are summed to find the total volume. To assist in setting up the integrals, draw lots of sketches!


By the end of this topic you should know and be prepared to be tested on:


Terms you should be able to define: axis of revolution, solid of revolution, slice, disk, washer, method of disks, method of washers, slicing methods, slab, cross-section, cross-section method

Mini-Lectures and Examples

STUDY: Geometry - Volume by Slicing and Shell Methods

Supplemental Resources (recommended)

Visualizing the disk/washer methods can be hard at first. Exploring these demonstrations may help:

Supplemental Resources (optional)

Video: Volume Calculations I (Cross Sections), Selwyn Hollis's Video Calculus

Video: Volume Calculations II (Disks/Washers), Selwyn Hollis's Video Calculus

Lesson: Solids of Revolution by Disks/Washers, Math is Fun (introductory information/examples)

Lesson: Volume with Rings, Paul's Online Notes for Calc I

Lesson: More Volumes, Paul's Online Notes for Calc I covers finding volume of non-revolution solids by cross-section method

PDF: Volumes of Solids, Dale Hoffman's Contemporary Calculus

Examples: Volume of a Solid with a Known Cross Section, Math24's Calculus includes several examples of finding volume of non-revolution solids by cross-section method, but watch that you only study the functions in x and y, not the parametric (in terms of t) nor polar (in terms of theta).

rev. 2021-01-16