Overview

Midpoints

Right Endpoints

Approximating Area

Riemann Sums is a method used to approximate the area between a graph and the x-axis. This method is often something you’ll study along with Trapezoidal Rule, Simpson’s Approximation, and the Fundamental Theorem of Calculus. All of these are different, but similar, ways to estimate the area underneath a curve. Each of these methods involve using basic geometric shapes, like rectangles and trapezoids, to get an estimate for the area under the curve. All of these techniques are stepping stones to your understanding of integration, which will allow you to find the exact area under the curve, instead of just approximate it.

Left, Right and Midpoints

Since we don’t know yet how to calculate the exact area under a curve (we’ll learn how to do this later with integration), we’ll need to use Riemann Sums to find an approximation of the area. The basic idea behind a Riemann Sum approximation is to use rectangles to approximate the area under the curve. The more rectangles we draw, the better our area approximation will be, as you can see below.

To use a Riemann Sum approximation, you’ll simply calculate the area of each rectangle under the graph, and then add the area of each rectangle together to get the total area under that section of the graph. The accuracy of the approximation will depend on whether we use the left-endpoints of the rectangles, the right-endpoints, or their midpoints. All three choices will give us different approximations of the area. Midpoint approximations are usually the most accurate.

Over and Underestimation

• If the function is increasing, left endpoints will give an underestimation and right endpoints give an overestimation.
• If the function is decreasing, left endpoints will give an overestimation and right endpoints give an underestimation.
• Midpoints will almost certainly give us a better approximation of both increasing and decreasing functions.
• If the function is both increasing and decreasing, it’s going to be difficult for us to determine whether or not we’ll get an under or overestimation from using left or right endpoints.

Calculating a Riemann Sum

In order to calculate a Riemann Sum, follow these steps:

1. Find delta x, where b is the upper limit of the range, a is the lower limit of the range, and n is the number of intervals you divide your range into.

2. Divide your range and mark off your x-axis into increments of delta x.

3. Decide whether you’ll use left endpoints, right endpoints, or midpoints of each of the rectangles indicated by the increments you created in Step 2.

4. Circle the values at which you’ll evaluate your function based on the legend below.

Left Endpoints

Right Endpoints

Midpoints

A Note About Integration

There is no limit to the number of rectangles you use to evaluate the area under the graph. The more rectangles you use, the more accurate your approximation will be. If you used an infinite number of rectangles, and therefore the value of delta x approached zero, you’d be integrating and finding the exact value of the area, instead of just an approximation.