## 7.1. Riemann Integral

### Definition 7.1.3: Riemann Sums

If

*P = { x*is a partition of the closed interval_{0}, x_{1}, x_{2}, ..., x_{n}}*[a, b]*and*f*is a function defined on that interval, then the**is defined as:***n*-th Riemann Sum of*f*with respect to the partition*P*whereR(f, P) = f(t_{j}) (x_{j}- x_{j-1})

*t*is an arbitrary number in the interval_{j}*[x*._{j-1}, x_{j}]
**Note:** If *t _{i}* is always the left endpoint of each
subinterval, the corresponding Riemann sum is called

*left Riemann sum*; if it is always the right endpoint, the sum is called

*right Riemann sum*.