## 7.1. Riemann Integral

### Examples 7.1.9(a):

Show that the constant function

We have to compute the upper and lower sums for an arbitrary
partition, then find the appropriate *f(x) = c*is Riemann integrable on any interval*[a, b]*and find the value of the integral.*inf*and

*sup*to compute the lower and upper integrals. If they agree, we are done and the common value is the answer. So, here we go:

Take an arbitrary partition
*P = {x _{0}, x_{1}, ..., x_{n}}*.
The lower sum of

*f(x) = c*is:

because theL(f, P) = c (x_{1}- x_{0}) + c (x_{2}- x_{1}) + ... + c (x_{n}- x_{n-1})

= c (x_{n}- x_{0}) = c (b - a)

*inf*over any interval (as well as the

*sup*) is always

*c*, and the above sum is telescoping.

Similarly, we have that

Hence, the upper and lower sums are independent of the particular partition. ThereforeU(f, P) = c (b - a)

*f*is integrable and

In particular,I^{*}(f) = I_{*}(f) = c (b - a)

f(x) dx = c (b - a)