Chapter 7 The Riemann Integral
The two pillars of calculus are differentiation and integration. Since we’ve tackled differentiation, let’s start on integration.
One thing to keep in mind is that integration is independent from differentiation -- that is, although we know and love the Fundamental Theorem of Calculus, we must realize that a definite integral is a thing in its own right.
So: what should the definite integral mean? Ideally, we’d define the definite integral of \(f\) on the interval \([a,b]\) to mean the area between the graph of \(f\text{,}\) the \(x\)-axis, and the lines \(x=a\) and \(x=b\text{.}\)
As we did with differentiation, we’ll first tackle this problem from the linear perspective. In this case, that means the function \(f\) is constant. So the area we’re talking about is that of a rectangle -- so the value of the definite integral of \(f(x)=c\) on \([a,b]\) is
\begin{equation*}
\displaystyle \int_{[a,b]}c\ dx = c(b-a)\ \ \ .
\end{equation*}
Our job will be to extend this definition in a sensible way.
