13.1 Integrals of Blips
We'll start our theory of integrals somewhere simple by looking at integrals of blips. The blip
\[
\begin{align}
[x_0]_{x} &: \Blip_{x}
\end{align}
\]
selects the point \( x_0 \) on the \( x \)-axis.
We can think of the blip as representing the equation \( x = x_0 \).
Definition.
Suppose we have a function and a blip:
\[
\begin{align}
z &\depends x,
&
b &: \Blip_{x}.
\end{align}
\]
The integral of \( b \)
\[
\begin{align}
\Bigl\langle{z}\mathrel{\textstyle{\int}}{b}\Bigr\rangle &: \Number
\end{align}
\]
is the number obtained by evaluating \( z \) at the blip \( b \).
Let's see how this works.
Example.
Let's compute the integral for the following function and blip.
\[
\begin{align}
z &\depends x
&
&
&
b &: \Blip_{x}
\\
z &= 3 x^2
&
&\andSpaced
&
b &= [2]_{x}
\end{align}
\]
This is a quick calculation.
\[
\begin{align}
\biggl\langle{z}\mathrel{\displaystyle{\int}}{b}\biggr\rangle &= \biggl\langle{3 x^2}\mathrel{\displaystyle{\int}}{[2]_{x}}\biggr\rangle = \biggl\langle{12}\mathrel{\displaystyle{\int}}{\sliced}\biggr\rangle = 12
\end{align}
\]
The blip \( [2]_{x} \) tells us to substitute two for \( x \). Having used up the blip, there is nothing left to be done, and so we've found our result.
A product of blips \( [x_0]_{x} \mult [y_0]_{y} \) allows us to select the point \( (x_0, y_0) \) in the coordinate plane.
We define an integral of blips for a function \( z \depends (x, y) \) in a similar manner.
Example.
Let's compute the following integral of blips.
\[
\begin{align}
\biggl\langle{\frac{x + y}{x - y}}\mathrel{\displaystyle{\int}}{[5]_{x} \mult [3]_{y}}\biggr\rangle &= \biggl\langle{\frac{5 + y}{5 - y}}\mathrel{\displaystyle{\int}}{[3]_{y}}\biggr\rangle = \biggl\langle{\frac{8}{2}}\mathrel{\displaystyle{\int}}{\sliced}\biggr\rangle = 4
\end{align}
\]
Here we applied the \( x \)-blip before the \( y \)-blip. We could equally well have applied the \( y \)-blip first.