5.3 The Definite Integral



When you finish this section, you should be able to:

  1. Define a definite integral as the limit of Riemann sums (p. 11)
  2. Find a definite integral using the limit of Riemann sums (p. 14)

The area \(A\) under the graph of \(y = f(x)\) from \(a\) to \(b\) is obtained by finding \[ \begin{equation} \boxed{\bbox[#FAF8ED,5pt]{{A=\lim\limits_{n\rightarrow \infty}s_{n}=\lim\limits_{n\rightarrow \infty }{\sum\limits_{i=1}^{n}} {f}(c_{i}) \Delta x=\lim\limits_{n\rightarrow \infty }S_{n}=\lim\limits_{n\rightarrow \infty }{\sum\limits_{i=1}^{n}} {f}(C_{i}) \Delta x }}} \end{equation} \]

THEOREM The Integral of the Sum of Two Functions

If two functions \(f\) and \(g\) are continuous on the closed interval \([a,b]\), then \[ \begin{equation*} \boxed{\bbox[5pt]{\int_{a}^{b}[ f(x)+g(x)] \,dx=\int_{a}^{b}f(x)\,dx+\int_{a}^{b}g(x)\,dx}}\tag{1} \end{equation*} \]