# Definite Integrals

### Example 1

Let *f* (*x*) = sin *x* + 1. Use a midpoint sum with 2 equal sub-intervals to estimate the area between *f* and the *x*-axis on [0,2π].

### Example 2

Let *g* (*x*) = *x*^{3} on the interval [0,12]. Use a midpoint sum with 4 equal sub-intervals to estimate the area between *g* and the *x*-axis on this interval.

### Example 3

Partial values of the function *h* are given in the table below. Use a midpoint sum with 3 intervals to estimate the area between *h* and the *x*-axis on [-2,4] or explain why you can't.

### Example 4

Partial values of the function *f* ( *x* ) are given in the table below. Use a midpoint sum with 4 equal sub-intervals to estimate the area between *f* and the *x*-axis on [0,1], or explain why you can't.