## 17.1 Signed Areas and Volumes

Area, like distance, and volume in customary language are quantities that
are always positive. However, we will find it useful to give signs to them.

Thus
if you are driving a car, and another car is 2 car lengths ahead of you,
you might assign a positive distance to the distance between your car
and it, and if it is behind you we can assign a negative distance to the same.

The
same sort of thing can be done with area and volume. If you have an x-axis,
you can assign positive area to area above it, and negative area to that
below it. And the same idea can be generalized to volumes, in three dimensions,
and
even further.

**Why would you want to do this?**

If you plot the distance between you and an
oncoming vehicle, when you are standing still, this distance will decrease
as it approaches, and then increase
again, after it goes past you. Thus the plot of its distance will look like
a V. If we use signed distance, and the vehicle is moving at a uniform speed,
the distance from where you were before you dived out of the way will be
a straight line. After it passes you its distance becomes negative. Straight
lines are so much easier to deal with than V-like curves that we prefer to
deal with them.

The area of a rectangle, as I hope you remember, is the product
of the lengths of its sides, if we ignore signs, which we normally do. This
is the basic fact
we start from.

Similarly, the volume of a cube is the cube of its side length.
The analogue of a rectangle in three dimensions is called a "rectangular
parallelepiped" and
its volume is the product of the lengths of its three sides. And you can imagine
similar statements in more dimensions.

We will now discuss the areas of tilted
parallelograms, and the volumes of general parallelepipeds, which are three
dimensional six sided figures whose
opposite sides are parallel to one another.

**Why?**

You will soon see why. Be patient and you will learn something you do
not now know.