ELi5: If the “rate of change” of a function is a tangible way to understand derivatives, what is a similar way to understand integrals?

Question

I know it’s the “area under the curve”, but what does that mean exactly? Is there a physical or tangible way to explain it?

I understand that a derivative is rate of change at a specific point, and something like acceleration is rate of change of speed. But how can I visualize that speed is the “integral” of acceleration? What does that mean, and how does it relate to the area underneath?

In: 172

Answer 1

I find velocity to be an easy example. The derivative of a velocity function is the acceleration function, the change in velocity. The integral of velocity is the position function, your current position over time.

One way to look at an integral is the definite integral. As opposed to the indefinite integral, which is just a function, a definite integral includes start and end points on the x axis (it can get more complicated but screw that). Solving a definite integral is what people refer to as “the area under the curve”, that is the area between your original function line and the x axis. For velocity, a definite integral of velocity is the total position change, or distance covered, between two times. For example, driving for an hour at 50mph constantly, you could take the definite integral of 50mph over the hour to get how much distance you covered, which happens to be 50 miles. Finding the “area under the curve” is easy for linear functions, as setting a start and end x value makes a polygon on the graph that you can just use geometry on to get the area, but you need the calculus magic to do this for more curvy functions like x squared.

Another way to look at a velocity integral is the current position at the given time. If you pick an x value on the velocity function, you can plug that x into the integral function to get the position at that time. Remember that constants are lost when differentiating, so the integral of a function is always some function “plus C”, the unknown constant. You can never know for sure what the exact position of something is if you only have the velocity function, but it will always differ from the true value by a constant “C” value. For velocity, the C value in the integral could be seen as your starting point. Maybe you’re driving across the country, but you start 10 miles in from one of the borders, so C may be “+10”.