Derivative and Integral: the play

[Background: Given a graph, I can visualize its derivative much more easily than I can visualize any of its antiderivatives. Time for that to change, even if I have to resort to something as ridiculous as giving those operations personalities.]
  
Suppose a graph represents how good the day is going versus time. Positive values mean a good day; negative values mean a bad day. A derivative is like a person who only cares about whether things are getting better or worse, and whose mood is based solely on that direction. It doesn't care about the past. It doesn't care about how good things actually are right now. The integral is like a person whose mood gets gradually better or worse depending solely on how good things are in the moment. It's affected very much by the past. It doesn't care if things are improving or worsening.

Let's graph these characters' moods over a day that has its up and downs, as represented by the graph of cosine. Because Integral's mood depends on his past mood, I pick an arbitrary starting mood for him at 0.

t=0

Integral: Wow, what a day, huh? This is tops!
Derivative: If you think so, why aren't you happier? Your happiness is at 0.
Integral: Oh, I'm getting happy very quickly! Look how quickly my mood is climbing! Besides, you're at 0 also.
Derivative: But I'm at 0 because the day has stagnated. Who knows what the future holds...
Integral: Pessimist!

t=0.6

Integral: Why so negative?
Derivative: It's all downhill from here. At this rate, we'll start having a bad day.
Integral: Just enjoy the present! It's still a great day!
Derivative: Well, I notice you're not getting happier quite as fast.
Integral: Because it's not quite as nice a day. But it's still pretty great!

t=1.2

Derivative: Wow, the day's really taken a turn for the worst, hasn't it? I'm so bummed.
Integral: You're so worried about the future that you can't enjoy the present. I'm getting happier all the time!
Derivative: Yes, but not nearly as quickly.
Integral: Who cares? The day's still good, and I'm still getting happier.

t=π/2 (about 1.6)

Integral: Well, it doesn't get better than this! I'm so happy!
Derivative: You do realize you're coasting, right? The day's no longer good, and you're not getting happier.
Integral: I'm high from all the good stuff that just happened. I can't understand why you're so depressed.
Derivative: Isn't it obvious? This is the fastest rate at which the day has declined!
Integral: Why do I even talk to you?

t=2.5

Integral: I'm still happy, but I have to admit the day's sucked lately. It's bringing me down.
Derivative: Well, I don't feel quite as bad. The day's not declining as quickly.

t=π (about 3.14)

Derivative: Did you notice? The day stopped getting worse. I'm no longer sad. If things start looking up, I'll actually be happy.
Integral: Yeah, well the last part of this day sucked so much, it wiped out all my happiness from the first part. I'm no longer happy. This could start to irritate me.

t=4

Derivative: Hey, the day's finally headed in the right direction. This is great!
Integral: Shut up. It sucks. I'm feeling worse and worse.
Derivative: But look at the trend! It's improving.
Integral: You're even more irritating when you're happy.

t=3π/2 (about 4.7)

Derivative: Wow, things are really looking up! I couldn't be happier!
Integral: We just came through so much suck. Twice as much suck as good stuff, in fact. This is misery.

t=5.5

Derivative: You have to admit, things have been good lately.
Integral: Yes, so well that it's pulling me out of this bad mood. But you're not as happy. 
Derivative: I'm still happy, just not quite as much, because things aren't looking up quite as much.
  

t=2π: repeat