Learning Objectives
 Use Riemann Sums to approximate areas under curves
 Evaluate definite integrals as limits of Riemann Sums
Introduction
In the Lesson The Area Problem we defined the area under a curve in terms of a limit of sums.
where
and were examples of Riemann Sums. In general, Riemann Sums are of form where each is the value we use to find the length of the rectangle in the subinterval. For example, we used the maximum function value in each subinterval to find the upper sums and the minimum function in each subinterval to find the lower sums. But since the function is continuous, we could have used any points within the subintervals to find the limit. Hence we can define the most general situation as follows:
 Definition
 If is continuous on we divide the interval into subintervals of equal width with . We let be the endpoints of these subintervals and let be any sample points in these subintervals. Then the definite integral of from to is
Example 1:
Evaluate the Riemann Sum for from to using subintervals and taking the sample points to be the midpoints of the subintervals.
Solution:
If we partition the interval into equal subintervals, then each subinterval will have length So we have and
Now let’s compute the definite integral using our definition and also some of our summation formulas.
Example 2:
Use the definition of the definite integral to evaluate
Solution:
Applying our definition, we need to find
We will use right endpoints to compute the integral. We first need to divide into subintervals of length Since we are using right endpoints,
So
Recall that By substitution, we have
Hence
Before we look to try some problems, let’s make a couple of observations. First, we will soon not need to rely on the summation formula and Riemann Sums for actual computation of definite integrals. We will develop several computational strategies in order to solve a variety of problems that come up. Second, the idea of definite integrals as approximating the area under a curve can be a bit confusing since we may sometimes get results that do not make sense when interpreted as areas. For example, if we were to compute the definite integral then due to the symmetry of about the origin, we would find that This is because for every sample point we also have is also a sample point with Hence, it is more accurate to say that gives us the net area between and If we wanted the total area bounded by the graph and the axis, then we would compute .
Lesson Summary
 We used Riemann Sums to approximate areas under curves.
 We evaluated definite integrals as limits of Riemann Sums.
Multimedia Link
For video presentations on calculating definite integrals using Riemann Sums (13.0), see Riemann Sums, Part 1 (6:15)
and Riemann Sums, Part 2 (8:31).
The following applet lets you explore Riemann Sums of any function. You can change the bounds and the number of partitions. Follow the examples given on the page, and then use the applet to explore on your own. Riemann Sums Applet. Note: On this page the author uses Left and Right hand sums. These are similar to the sums and that you have learned, particularly in the case of an increasing (or decreasing) function. Lefthand and Righthand sums are frequently used in calculations of numerical integrals because it is easy to find the left and right endpoints of each interval, and much more difficult to find the max/min of the function on each interval. The difference is not always important from a numerical approximation standpoint; as you increase the number of partitions, you should see the Lefthand and Righthand sums converging to the same value. Try this in the applet to see for yourself.
Review Questions
In problems #1–7 , use Riemann Sums to approximate the areas under the curves.
 Consider from to Use Riemann Sums with four subintervals of equal lengths. Choose the midpoints of each subinterval as the sample points.
 Repeat problem #1 using geometry to calculate the exact area of the region under the graph of from to (Hint: Sketch a graph of the region and see if you can compute its area using area measurement formulas from geometry.)
 Repeat problem #1 using the definition of the definite integral to calculate the exact area of the region under the graph of from to
 from to Use Riemann Sums with five subintervals of equal lengths. Choose the left endpoint of each subinterval as the sample points.
 Repeat problem #4 using the definition of the definite intergal to calculate the exact area of the region under the graph of from to
 Consider Compute the Riemann Sum of on under each of the following situations. In each case, use the right endpoint as the sample points.
 Two subintervals of equal length.
 Five subintervals of equal length.
 Ten subintervals of equal length.
 Based on your answers above, try to guess the exact area under the graph of on
 Consider . Compute the Riemann Sum of on under each of the following situations. In each case, use the right endpoint as the sample points.
 Two subintervals of equal length.
 Five subintervals of equal length.
 Ten subintervals of equal length.
 Based on your answers above, try to guess the exact area under the graph of on
 Find the net area under the graph of ; to (Hint: Sketch the graph and check for symmetry.)
 Find the total area bounded by the graph of and the axis, from to to
 Use your knowledge of geometry to evaluate the following definite integral: (Hint: set and square both sides to see if you can recognize the region from geometry.)
Review Answers

 The graph is symmetric about the origin; hence
 The graph is that of a quarter circle of radius ; hence