# Untitled Page 38

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## 9.4 Spherical and cylindrical coordinates

In cylindrical coordinates (*R*,φ,*z*),
*z* measures distance
along the axis, *R* measures distance from the axis, and φ
is an angle that wraps around the axis.

The differential of volume in cylindrical coordinates can be
written as *dv* = *R* *dR* *dz* d φ. This follows from adding a third dimension,
along the *z* axis, to the rectangle in figure d.

##### Example 7

◊ Show that the expression for *dv* has the right units.

◊ Angles are unitless, since the definition of radian measure involves a distance
divided by a distance. Therefore the only factors in the expression that have units
are *R*, *dR*, and *dz*. If these three factors are measured, say, in meters, then
their product has units of cubic meters, which is correct for a volume.

##### Example 8

◊ Find the volume of a cone whose height is *h* and whose base has radius *b*.

◊ Let's plan on putting the *z* integral on the outside of the sandwich. That means
we need to express the radius *r*_{max} of the cone in terms of *z*. This comes out nice and
simple if we imagine the cone upside down, with its tip at the origin.
Then since we have *r*_{max}(*z*=0)=0, and *r*_{max}(*h*)=*b*, evidently *r*_{max}=*zb*/*h*.

As a check, we note that the answer has units of volume. This is the classical result, known by the ancient Egyptians, that a cone has one third the volume of its enclosing cylinder.

In spherical coordinates
(*r*,θ,φ), the coordinate
*r* measures the distance from the origin, and θ and φ are analogous to
latitude and longitude, except that θ is measured down from the pole rather than
from the equator.

The differential of volume in spherical coordinates is *dv* = *r*^{2}sinθ *dr*dθdφ.

##### Example 9

◊ Find the volume of a sphere.

◊