Let be a continuous function, not necessarily nonnegative. Partition into consecutive sub-intervals () each of length , where we set , and to be successive points between and with . Let be any intermediate point in the sub-interval . Then the sum

is called a Riemann sum for on .

Suppose we let the number of partition in tends to infinity.

We call the Riemann integral (or definite integral) of over and we write

In other words,

if the limit on the right side exists.

If we put we the obtain

**Example 1**. Find

*Solution*. Clearly

Then if we choose , we then get

With this it is easy to see that

since

If we put we the obtain

**Example 2**. Find

*Solution*. Clearly,

which yields

**Remark**. It is worth mentioning that in general it is not true that

For example, we all know that for each fixed

but

The point is

holds true only for finite summation.