Quandaries and Queries
Who is asking: Student
Level of the question: Secondary
Question: When taking the integral of the position function, you get the velocity function, and the same for velocity to acceleration. So when you do each of these, you get a function. But when you integrate on a graph, you get an area under a curve. The area is un units squared- where do the units go when you make it an equation? How can a function be an area?
You have integration confused with differentiation. If you take the derivative of the position function with respect to time you get the velocity function. Likewise if you take the derivative of the velocity function with respect to time you get the acceleration function.
Graphically this makes sense. The derivative gives you the slope of the tangent or the rate of change of the position with respect to time. So if you are measuring position in feet and time in seconds then slope of the tangent is the rate of change of position with respect to time in the units of feet per second.
Integration of the velocity function will give you the position function but in this instance you should think of integration as antidifferentiation. That is I give you a velocity function and ask you to tell me what function you would differentiate to get the function I gave you. This is the position function.
The connection between antidifferentiation and the calculation of areas is not obvious. It's called The Fundamental Theorem of Calculus.