In many problems, it is desired to reverse the process of differentiation and retrieve functions based on knowledge of their derivatives. This chapter introduces integration, the process of finding a function from its derivative.
In physics, the acceleration (a(t) = v'(t) = s''(t)) of a moving body as a function of time is often known (through dividing the net force exerted on the body by its mass). Knowing the current location and velocity of the body, we often want to calculate its future velocity and location. Specifically, if we consider a falling body, its acceleration, if the air resistance is negligible, is a constant where or . So how can we use this to calculate the velocity and the location of the object? Another example is when we need to find after how long radioactive wastes become harmless knowing the decay rate.
In such problems, we want to find a function whose derivative is given. If such a function exists, it is called an antiderivative (or integral) of , and the process of finding it is called "integration."
Recovering a function from its derivative is an important part of integral calculus. In this chapter, we study this meaning of integration. However, integration has another meaning which is close to its nontechnical meaning "to put or bring together (parts or elements) so as to form one whole; to combine into a whole" (Oxford English Dictionary). The second meaning of integration will be studied in the next chapter.
