Signal convolution: continuous signals

Convolution is comprised of three steps:

• Introduce a dummy variable $tau$ and use it to represent our functions. Also, reflect a function ($x(tau)$) about $x=0$ with our dummy variable: $x(-tau)$.
• Introduce a time offset for that function ($t$) allowing us to 'slide' $x(-tau)$ along the $x$ axis.
• Find the integral of the product of our two functions at key values of $t$.

So, let's reflect $x(t)$ by making it $x(-tau)$. You'll note that at this point, neither of the functions are overlapping and therefore the integral of their product is 0. Now, we're going to use $t$ to slide $x(-tau)$ toward $xrightarrow infty$ and it will begin to overlap with $h(tau)$: $x(t-tau)$.

At $t=4$ the two rectangular pulses will be half-overlapping eachother. Therefore, the integral of their product is going to be $4times20=80$.

At $t=8$ the two rectangular pulses will be completely overlapping eachother. They're symmetrical so the integral of the product is now going to be $8times20=160$.

For $t>8$ the two rectangular pulses will begin to move away from one another and thus the integral of their product will begin to decline.

A plotted result of this convolution would look like: