## 7.1. Riemann Integral

### Definition 7.1.5: Upper and Lower Sum

Let

*P = { x*be a partition of the closed interval_{0}, x_{1}, x_{2}, ..., x_{n}}*[a, b]*and*f*a bounded function defined on that interval. Then:
The **upper sum** of *f* with respect to the
partition *P* is defined as:

whereU(f, P) = c_{j}(x_{j}- x_{j-1})

*c*is the supremum of_{j}*f(x)*in the interval*[x*._{j-1}, x_{j}]
The **lower sum** of *f* with respect to the
partition *P* is defined as

whereL(f, P) = d_{j}(x_{j}- x_{j-1})

*d*is the infimum of_{j}*f(x)*in the interval*[x*._{j-1}, x_{j}]