Chapter 6 Integration
As mentioned in the Differentiation chapter, the derivative was developed to solve the “instantaneous velocity” problem, or “the slope of the tangent line” problem. The integral has instead origins in the area problem: what is the area of a curved region, such as the area under the graph of a given function? While the derivative has a simple definition as the limit of the difference quotient, the integral is harder to define. A crucial breakthrough in the late 1600’s (Newton, Leibniz) was the realization that the area problem can be solved as the “inverse” of the “slope of tangent line” problem: if we define \(F(x)\) to be the area under the graph of \(f\) from a fixed point \(a\) to a variable point \(x\text{,}\) then the Fundamental Theorem of Calculus says that \(F'(x)=f(x)\text{.}\) So solving the area problem is reduced to solving the “antiderivative problem”: for a given \(f\text{,}\) find a function \(F\) such that \(F'(x)=f(x)\text{.}\) As a consequence, for many years people focused on derivatives and antiderivatives and it was not until the 1800’s that a precise definition of the definite integral was given by B. Riemann. In this chapter, we will first give the original definition of the Riemann integral, then we will give a second definition from which it is easier to prove theorems. The two definitions can be proved to be equivalent.
