**Definite Integral:** If \(f\) is a continuous function defined on the interval \([a, b]\) that is divided into \(n\) equal subintervals \(\Delta x = \frac{{b - a}}{n}\) where \(x_i^*\) lies in the *i*th subinterval then the definite integral of \(f\) from \(a\) to \(b\) follows:

\[\int_a^b {f\left( x \right)dx} = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {f\left( {x_i^*} \right)} \,\Delta x\]

**Instructions:**Drag the slider to see that the area under the curve is the limit of Riemann sums. You can also change the sums to left, middle, right, and trapezoidal below the graph.

Riemann Sum:
Left
Middle
Right
Trapezoidal