This is a recap of how we set up double integrals, using the example
of finding the volume under *z* = 2 *y* between
the cylinders
*x*^{2} +
*y*^{2} = 1
and
*x*^{2} +
*y*^{2} = 4.
First, recall that our steps are the following:

To set up a double integral over a region,

,
we go through the three steps

- Sketch the region,
- Consider one variable as fixed at successive values across the
region, and determine the range of values it can assume when
doing this -- this gives the
*outer limits* in the
double integral, which *must be constants*,
- At each of the possible values for the outer variable,
determine the corresponding values for the other, inner
variable.

Doing the third step requires us thinking about what, geometrically,
fixing one variable means: if we fix

*x* or

*y*, this means
that we're restricting ourselves to a vertical or horizontal line at
some

*x* or

*y* value. If we fix

*r* in

*polar
coordinates*, it means we're restricted a circle of that radius
centered at the origin, and if we fix

*theta* in polar
coordinates, we're restricted to a line through the origin at an angle

*theta* from the

*x*-axis.

Let's consider this for the example given above, recapped here for
your viewing pleasure:

Find the volume under *z* = 2 *y* between
the cylinders
*x*^{2} +
*y*^{2} = 1
and
*x*^{2} +
*y*^{2} = 4.

We're going to approach this two ways, setting it up both in
rectangular and polar coordinates. We'll illustrate the steps given
above with a figure, too.