Slicing and Dicing Solids

Main idea:  To find the volume of a solid, we slice the solid into thin pieces and add up the volumes of the pieces. 

When we slice a solid, we get cross-sections of the solid.  For example, when we slice a solid cylinder, we see that we can get cross-sections that are rectangles or circles, depending upon how we slice.  

                    
 http://calculus.seas.upenn.edu/ 

In general, we do not have nice solids like a cylinder, but we can slice any solid.  We put our solid on axes, so that we can coordinatize our slices, as in the diagrams below.
                                                                                                                                                                              http://www2.bc.cc.ca.us/                                                        http://www.emathhelp.net

We slice the solid into these cross-sectional pieces, then add up the volumes of each of the pieces, to get the volume of the solid.  If a slice has cross-sectional area $A(x)$ and thickness $\Delta x$ at each particular $x$-value, then the slice has approximate volume $A(x) \Delta x$. Adding up the volumes of the slices and taking a limit as we slice finer and finer, we get

where $a$ and $b$ are the $x$-values of the leftmost and rightmost slices.

Of course, we could also slice horizontally,
                                                                         

                                                                                 http://calculus.seas.upenn.edu/                         http://tutorial.math.lamar.edu/
in which case we would find the volume to be $\int_c^d A(y)\,dy$, where $c$ is the smallest $y$-value, and $d$ is the largest ($c=0,d=h$ in the second diagram above).

The video will make these ideas more concrete.