Everybody knows that the voluem of a solid box is
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
The volume of our potentially complicated blob
cross sectional area at $x$
|volume of blob
Find the volume of the pyramid with height
square base with sides of length
How Big is Pharaoh's Place?
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
. The equation of this line is thus
. Thus the cross sectional area is
The volume is then
Next: Polar coordinates, etc.
Recall: Find volume by integrating cross section of area. (draw picture)
Find the volume of the solid obtained by rotating the following
region about the
axis: the region enclosed by
Find the volume of the flower pot
The cross section is a ``washer'', and the area as a function
The volume is thus
One of the most important examples of a volume is the volume
of a sphere of radius
. Let's find it!
We'll just compute the volume of a half and multiply by
Cross section of a half of sphere with radius 1
The cross sectional area is
Find volume of intersection of two spheres of radius
the center of each sphere lies on the edge of the other sphere.
From the picture we see that the answer is
as in Example 3.2.3