You have done a good job. A simple example helps any student to see easily what is happening before they meet much more complicated examples. I like the way you have highlighted the text as you speak. This must be a great help to any for whom English is not the first language.
Defining integration as a summation process is the way I would do it too. It is possible to prove that the area under a curve gives the same result so I would see this as a consequence of integration rather than another way of defining it. When I was at school my maths teacher said the proof was too hard and 'glossed' over it. But I looked it up and I don't think it is that tough.
Children are not defined by school ...........The Fonz
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
Thanks indeed mr.Bob for this neat comment. I believe that Visual materials can abstract thousands of words and a lot of time!
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.