If you want to understand a real-world problem, draw a picture of it.
By graphing a function, you turn hard-to-understand questions about physics
or economics or anything else into questions of geometry. In particular:
The slope of a graph is related to how fast the quantity being
graphed is changing. Although you never walk down the street and wonder
"what's the slope of that curve?", you often need to know how fast something
(like the position of the car bearing down on you) is changing.
Mathematically, they're almost the same thing.
The area under a curve tells you how much of the stuff being
graphed has accumulated. If the graph is about income, then the area is lifetime
earnings. If the graph is about velocity, then the area is distance
traveled.