volume length width height

More generally, the volume of cylinder is
(cross sectional area times height).
Even more generally, if the base of a prism has area , the
volume of the prism is .
But what if our solid object looks like a complicated blob? How would we compute the volume? We'll do something that by now should seem familiar, which is to chop the object into small pieces and take the limit of approximations.

[[Picture of solid sliced vertically into a bunch of vertical thin solid discs.]]

Assume that we have a function

cross sectional area at $x$

The volume of our potentially complicated blob
is approximately
.
Thus
volume of blob | ||

For convenience look at pyramid on its side, with the tip of the pyramid at the origin. We need to figure out the cross sectional area as a function of , for . The function that gives the distance from the axis to the edge is a line, with and . The equation of this line is thus . Thus the cross sectional area is

Today: Quiz!
Next: Polar coordinates, etc. Questions:? Recall: Find volume by integrating cross section of area. (draw picture) |

The cross section is a ``washer'', and the area as a function of is

From the picture we see that the answer is

William Stein 2006-03-15