Consider the following question:

You are driving in a car.

You know how fast you have been driving for the past hour.

You want to know: how far have you driven?

If your speed has been constant =s (mph), the answer is:

distance travelled = (speed).(time) = s

Otherwise you can ide the time interval into segments, estimate your speed in each segment; compute your distance travelled in it, and add these up.

The distance travelled in a time interval is the definite integral of the speed of motion with respect to time. And this calculation is exactly a Riemann sum for this integral. 

If your time segments are small enough that your speed in each of them is essentially constant, your sum will be very accurate.

In practice you might use an easier method. You might total up and estimate the amount of time you  drove at each of several representative speeds, and add the distance travelled at them:   

Suppose you drove for five minutes in traffic at about 30 mph then half an hour on a highway at about 64 mph; stopped for five minutes then drove fifteen more minutes on the highway and finally five more minutes in traffic, with speeds as before. Then you might say:

My total distance travelled was near 64 * 3/4 + 30/6 or 53 miles.

You are then exploiting the fact that here only the length of time driven at a speed is important, not the distribution of that time on the time axis.   

The area and definite integral can be defined in general in this same way:

1. Divide the y axis into n intervals of size

2. Let y_k be the midpoint of the kth interval.

Estimate the area as (total length of all x interval such that f(x) lies in the kth y interval).

3. Take the limit as n approaches infinity.

Comment

If this limit exists, it is called the Lebesgue integral.