Approximating Areas with Riemann Sums

#6.2 Approximating Areas with Riemann Sums

Welcome back to AP Calc with Fiveable! In the last guide, we explored accumulations of change and thought about how to compute them. Here, we’ll take a deep dive into Riemann sums, a method for approximating the area under the curve to find the accumulation of change.

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#📶 Graphical Riemann Sums

As we delve into integral calculus, our goal has shifted from computing the instantaneous rate of change to computing the area under the curve. Riemann sums are a useful tool for approximating this. Take a look at this graph:

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Graph of a positive, increasing, concave down function.

Image courtesy of Emery

You would have a hard time computing this geometrically. But, we can approximate it using familiar shapes, like this:

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Same graph with 6 left-endpoint Riemann rectangles overlaid.

Image courtesy of Emery

We can see that this isn’t exact, but that our approximation improves if we use more and more rectangles:

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The same graph with double and quadruple (12 and 24) left-endpoint Riemann rectangles overlaid.

Image courtesy of Emery

The use of these rectangles to approximate the area under the curve is called a Riemann sum. There are four main Riemann sums:

1. Left Riemann Sum
2. Right Riemann Sum
3. Midpoint Riemann Sum
4. Trapezoidal Sum

Now, let’s get into each of these in detail!

#↔️ Left and Right Riemann Sum

There are two basic types of Riemann sums, called “left endpoint” and “right endpoint.” Here is an example of the same curve with a left Riemann sum, versus one with a right Riemann sum:

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The same graph with left- and right-endpoint Riemann rectangles overlaid.

Image courtesy of Emery

You can see that the left and right refer to which points we use to determine the height of our rectangles. Left Riemann sums touch the curve with their top left corners, and right Riemann sums touch the curve with their top right corners.

#➗ Subdivisions

Another thing that we can vary when deciding to take a Riemann sum is the width of our base. Our base length can either be uniform or non-uniform, as illustrated here:

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The same graph with uniform and non-uniform subdivisions of right Riemann rectangles overlaid.

Image courtesy of Emery

The base lengths formed by dividing up the total interval are called “subdivisions.”

#🤔 Overestimating vs Underestimating

We know that a Riemann sum isn’t a perfect calculation of our area under the curve, but are we overestimating or underestimating when we use one? Try these problems and then fill in the blanks to see if you can figure out the pattern!

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A set of four graphs. Two are increasing and two are decreasing. Demonstrates the relationship between increasing/decreasing and whether a right or left Riemann sum is an over- or underestimate.

Image courtesy of Emery

Fill in the Blanks! 🧠

• A left Riemann sum ________when a function is ________.
• A right Riemann sum ________w...