1.6: Definition of the Integral

The definite integral of a function \(f(x) > 0\) from \(x = a\) to \(b\) \((b > a)\) is defined as the area bounded by the vertical lines \(x = a,\: x = b\), the \(x\)-axis and the curve \(y = f(x)\). This “area under the curve” is obtained by a limit. First, the area is approximated by a sum of rectangle areas. Second, the integral is defined to be the limit of the rectangle areas as the width of each individual rectangle goes to zero and the number of rectangles goes to infinity. This resulting infinite sum is called a Riemann Sum, and we define \[\label{eq:1}\int_a^b f(x)dx=\underset{h\to 0}{\lim}\sum\limits_{n=1}^{N} f(a+(n-1)h)\cdot h,\] where \(N = (b − a)/h\) is the number of terms in the sum. The symbols on the lefthand-side of \(\eqref{eq:1}\) are read as “the integral from \(a\) to \(b\) of \(f\) of \(x\) dee \(x\).” The Riemann Sum definition is extended to all values of \(a\) and \(b\) and for all values of \(f(x)\) (positive and negative). Accordingly, \[\int_b^a f(x)dx=-\int_a^b f(x)dx\quad\text{and}\quad\int_a^b(-f(x))dx=-\int_a^bf(x)dx.\nonumber\]

Also, \[\int_a^c f(x)dx=\int_a^bf(x)dx+\int_b^c f(x)dx,\nonumber\] which states when \(f(x) > 0\) and \(a < b < c\) that the total area is equal to the sum of its parts