Here's what happens...
∫ f(x) dx (aka indefinite integral) means find all functions F(x) so that F'(x)=f(x).(Why does it use that funny symbol? Why does it have that dx part at the end? Hard to explain without referencing a connection that hasn't been made yet, isn't it?)
∫12 f(x) dx (aka definite integral) means find the area which is between f(x) and the x-axis (area below the axis counts as negative), and between x=1 and x=2.
Seeing almost identical notation and names, we're going to assume that these two act the same in some ways. Students are going to expect anti-derivatives, even though it's area we're talking about here. So it's not much surprise when the Fundamental Theorem of Calculus tells us that to find area we can use anti-derivatives.
Wait! That should be a surprise. It's kind of amazing, isn't it? Derivatives give us slope. Why would going backwards in that process give us area?! Seems to me that's a big one we need to meditate on for a while.
This semester I knew I wanted to connect the new ideas with the position, velocity, and acceleration problems, so I introduced anti-derivatives first. And, I showed the indefinite integral symbol. Oops! I shouldn't have. If I had held off, I believe the meaning of the definite integral would have taken hold better in my student's minds.
Until this semester, I've followed the textbook pretty closely, so my way around this problem has been to introduce the 'Area Function' without using this notation. I found this idea/project in a book put out by the MAA. I've revised it a lot over the years, but the original author, Charles Jones (of Grinnell College) still deserves credit for getting me started in this direction. (I wish I could figure out how to thank him personally, but he doesn't seem to be at Grinnell College these days, and google gives me lots of people with that name.)
I've put a pdf of the project here. If you'd like my Word file, just email me (mathanthologyeditor on gmail).
We've started that project, and it's going well enough, but I realized that if I hadn't introduced the indefinite integral, we'd be better off. Next semester I'll get that right.
Tomorrow we wrap up the project, and I clarify the implications of the Fundamental Theorem. Cool stuff!