4.2: Riemann Sums

For instance, for the velocity function given in Figure $4.2.1$, if the areas of shaded regions are $A_1\text{,}$ $A_2\text{,}$ and $A_3$ as labeled, then the total distance $D$ traveled by the moving object on $[a,b]$ is

$$D = A_1 + A_2 + A_3\text{,}$$

Sigma Notation

We have used sums of areas of rectangles to approximate the area under a curve. Intuitively, we expect that using a larger number of thinner rectangles will provide a better estimate for the area. Consequently, we anticipate dealing with sums of a large number of terms. To do so, we introduce sigma notation, named for the Greek letter $\Sigma\text{,}$ which is the capital letter $S$ in the Greek alphabet.

For example, say we are interested in the sum

the sum of the first 100 natural numbers. In sigma notation we write

We read the symbol $\sum_{k=1}^{100} k$ as “the sum from $k$ equals 1 to 100 of $k\text{.}$” The variable $k$ is called the index of summation, and any letter can be used for this variable. The pattern in the terms of the sum is denoted by a function of the index; for example,

and more generally,

Sigma notation allows us to vary easily the function being used to describe the terms in the sum, and to adjust the number of terms in the sum simply by changing the value of $n\text{.}$ We test our understanding of this new notation in the following activity.

Activity $4.2.2$

For each sum written in sigma notation, write the sum long-hand and evaluate the sum to find its value. For each sum written in expanded form, write the sum in sigma notation.

1. $\displaystyle \sum_{k=1}^{5} (k^2 + 2)$
2. $\displaystyle \sum_{i=3}^{6} (2i-1)$
3. $\displaystyle 3 + 7 + 11 + 15 + \cdots + 27$
4. $\displaystyle 4 + 8 + 16 + 32 + \cdots + 256$
5. $\displaystyle \sum_{i=1}^{6} \frac{1}{2^i}$

When a moving body has a positive velocity function $y = v(t)$ on a given interval $[a,b]\text{,}$ the area under the curve over the interval gives the total distance the body travels on $[a,b]\text{.}$ We are also interested in finding the exact area bounded by $y = f(x)$ on an interval $[a,b]\text{,}$ regardless of the meaning or context of the function $f\text{.}$ For now, we continue to focus on finding an accurate estimate of this area by using a sum of the areas of rectangles. Unless otherwise indicated, we assume that $f$ is continuous and non-negative on $[a,b]\text{.}$

The first choice we make in such an approximation is the number of rectangles.

If we desire $n$ rectangles of equal width to subdivide the interval $[a,b]\text{,}$ then each rectangle must have width $\Delta x = \frac{b-a}{n}\text{.}$ We let $x_0 = a\text{,}$ $x_n = b\text{,}$ and define $x_{i} = a + i\Delta x\text{,}$ so that $x_1 = x_0 + \Delta x\text{,}$ $x_2 = x_0 + 2 \Delta x\text{,}$ and so on, as pictured in Figure $4.2.1$3.

We use each subinterval $[x_i, x_{i+1}]$ as the base of a rectangle, and next choose the height of the rectangle on that subinterval. There are three standard choices: we can use the left endpoint of each subinterval, the right endpoint of each subinterval, or the midpoint of each. These are precisely the options encountered in Preview Activity $4.2.1$1 and seen in Figure $4.2.1$2. We next explore how these choices can be described in sigma notation.

Consider an arbitrary positive function $f$ on $[a,b]$ with the interval subdivided as shown in Figure $4.2.1$3, and choose to use left endpoints. Then on each interval $[x_{i}, x_{i+1}]\text{,}$ the area of the rectangle formed is given by

as seen in Figure $4.2.4$.

If we let $L_n$ denote the sum of the areas of these rectangles, we see that

In the more compact sigma notation, we have

Note that since the index of summation begins at $0$ and ends at $n-1\text{,}$ there are indeed $n$ terms in this sum. We call $L_n$ the left Riemann sum for the function $f$ on the interval $[a,b]\text{.}$

To see how the Riemann sums for right endpoints and midpoints are constructed, we consider Figure $4.2.5$.

For the sum with right endpoints, we see that the area of the rectangle on an arbitrary interval $[x_i, x_{i+1}]$ is given by $B_{i+1} = f(x_{i+1}) \cdot \Delta x\text{,}$ and that the sum of all such areas of rectangles is given by

We call $R_n$ the right Riemann sum for the function $f$ on the interval $[a,b]\text{.}$

For the sum that uses midpoints, we introduce the notation

so that $\overline{x}_{i+1}$ is the midpoint of the interval $[x_i, x_{i+1}]\text{.}$ For instance, for the rectangle with area $C_1$ in Figure $4.2.1$5, we now have

Hence, the sum of all the areas of rectangles that use midpoints is

and we say that $M_n$ is the middle Riemann sum for $f$ on $[a,b]\text{.}$

Thus, we have two variables to explore: the number of rectangles and the height of each rectangle. We can explore these choices dynamically, and the applet¹ found at http://gvsu.edu/s/a9 is a particularly useful one. There we see the image shown in Figure $4.2.6$, but with the opportunity to adjust the slider bars for the heights and the number of rectangles.

 

By moving the sliders, we can see how the heights of the rectangles change as we consider left endpoints, midpoints, and right endpoints, as well as the impact that a larger number of narrower rectangles has on the approximation of the exact area bounded by the function and the horizontal axis.

When $f(x) \ge 0$ on $[a,b]\text{,}$ each of the Riemann sums $L_n\text{,}$ $R_n\text{,}$ and $M_n$ provides an estimate of the area under the curve $y = f(x)$ over the interval $[a,b]\text{.}$ We also recall that in the context of a nonnegative velocity function $y = v(t)\text{,}$ the corresponding Riemann sums approximate the distance traveled on $[a,b]$ by a moving object with velocity function $v\text{.}$

There is a more general way to think of Riemann sums, and that is to allow any choice of where the function is evaluated to determine the rectangle heights. Rather than saying we'll always choose left endpoints, or always choose midpoints, we simply say that a point $x_{i+1}^*$ will be selected at random in the interval $[x_i, x_{i+1}]$ (so that $x_i \le x_{i+1}^* \le x_{i+1}$). The Riemann sum is then given by

At http://gvsu.edu/s/a9, the applet noted earlier and referenced in Figure $4.2.6$, by unchecking the “relative” box at the top left, and instead checking “random,” we can easily explore the effect of using random point locations in subintervals on a Riemann sum. In computational practice, we most often use $L_n\text{,}$ $R_n\text{,}$ or $M_n\text{,}$ while the random Riemann sum is useful in theoretical discussions. In the following activity, we investigate several different Riemann sums for a particular velocity function.

Activity $4.2.3$

Suppose that an object moving along a straight line path has its velocity in feet per second at time $t$ in seconds given by $v(t) = \frac{2}{9}(t-3)^2 + 2\text{.}$

1. Carefully sketch the region whose exact area will tell you the value of the distance the object traveled on the time interval $2 \le t \le 5\text{.}$
2. Estimate the distance traveled on $[2,5]$ by computing $L_4\text{,}$ $R_4\text{,}$ and $M_4\text{.}$
3. Does averaging $L_4$ and $R_4$ result in the same value as $M_4\text{?}$ If not, what do you think the average of $L_4$ and $R_4$ measures?
4. For this question, think about an arbitrary function $f\text{,}$ rather than the particular function $v$ given above. If $f$ is positive and increasing on $[a,b]\text{,}$ will $L_n$ over-estimate or under-estimate the exact area under $f$ on $[a,b]\text{?}$ Will $R_n$ over- or under-estimate the exact area under $f$ on $[a,b]\text{?}$ Explain.

When the function is sometimes negative

For a Riemann sum such as

we can of course compute the sum even when $f$ takes on negative values. We know that when $f$ is positive on $[a,b]\text{,}$ a Riemann sum estimates the area bounded between $f$ and the horizontal axis over the interval.

For the function pictured in the first graph of Figure $4.2.7$, a left Riemann sum with 12 subintervals over $[a,d]$ is shown. The function is negative on the interval $b \le x \le c\text{,}$ so at the four left endpoints that fall in $[b,c]\text{,}$ the terms $f(x_i) \Delta x$ are negative. This means that those four terms in the Riemann sum produce an estimate of the opposite of the area bounded by $y = f(x)$ and the $x$-axis on $[b,c]\text{.}$

In the middle graph of Figure $4.2.7$, we see that by increasing the number of rectangles the approximation of the area (or the opposite of the area) bounded by the curve appears to improve.

In general, any Riemann sum of a continuous function $f$ on an interval $[a,b]$ approximates the difference between the area that lies above the horizontal axis on $[a,b]$ and under $f$ and the area that lies below the horizontal axis on $[a,b]$ and above $f\text{.}$ In the notation of Figure $4.2.7$, we may say that

where $L_{24}$ is the left Riemann sum using 24 subintervals shown in the middle graph. $A_1$ and $A_3$ are the areas of the regions where $f$ is positive, and $A_2$ is the area where $f$ is negative. We will call the quantity $A_1 - A_2 + A_3$ the net signed area bounded by $f$ over the interval $[a,d]\text{,}$ where by the phrase “signed area” we indicate that we are attaching a minus sign to the areas of regions that fall below the horizontal axis.

Finally, we recall that if the function $f$ represents the velocity of a moving object, the sum of the areas bounded by the curve tells us the total distance traveled over the relevant time interval, while the net signed area bounded by the curve computes the object's change in position on the interval.

Activity $4.2.4$

Suppose that an object moving along a straight line path has its velocity $v$ (in feet per second) at time $t$ (in seconds) given by

$$v(t) = \frac{1}{2}t^2 - 3t + \frac{7}{2}\text{.}$$

1. Compute $M_5\text{,}$ the middle Riemann sum, for $v$ on the time interval $[1,5]\text{.}$ Be sure to clearly identify the value of $\Delta t$ as well as the locations of $t_0\text{,}$ $t_1\text{,}$ $\cdots\text{,}$ $t_5\text{.}$ In addition, provide a careful sketch of the function and the corresponding rectangles that are being used in the sum.
2. Building on your work in (a), estimate the total change in position of the object on the interval $[1,5]\text{.}$
3. Building on your work in (a) and (b), estimate the total distance traveled by the object on $[1,5]\text{.}$
4. Use appropriate computing technology² to compute $M_{10}$ and $M_{20}\text{.}$ What exact value do you think the middle sum eventually approaches as $n$ increases without bound? What does that number represent in the physical context of the overall problem?

For instance, consider the applet at http://gvsu.edu/s/a9 and change the function and adjust the locations of the blue points that represent the interval endpoints $a$ and $b\text{.}$

Summary

• A Riemann sum is simply a sum of products of the form $f(x_i^*) \Delta x$ that estimates the area between a positive function and the horizontal axis over a given interval. If the function is sometimes negative on the interval, the Riemann sum estimates the difference between the areas that lie above the horizontal axis and those that lie below the axis.
• The three most common types of Riemann sums are left, right, and middle sums, but we can also work with a more general Riemann sum. The only difference among these sums is the location of the point at which the function is evaluated to determine the height of the rectangle whose area is being computed. For a left Riemann sum, we evaluate the function at the left endpoint of each subinterval, while for right and middle sums, we use right endpoints and midpoints, respectively.
• The left, right, and middle Riemann sums are denoted $L_n\text{,}$ $R_n\text{,}$ and $M_n\text{,}$ with formulas
  $$\begin{align*} L_n = f(x_0) \Delta x + f(x_1) \Delta x + \cdots + f(x_{n-1}) \Delta x &= \sum_{i = 0}^{n-1} f(x_i) \Delta x,\\[4pt] R_n = f(x_1) \Delta x + f(x_2) \Delta x + \cdots + f(x_{n}) \Delta x &= \sum_{i = 1}^{n} f(x_i) \Delta x,\\[4pt] M_n = f(\overline{x}_1) \Delta x + f(\overline{x}_2) \Delta x + \cdots + f(\overline{x}_{n}) \Delta x &= \sum_{i = 1}^{n} f(\overline{x}_i) \Delta x\text{,} \end{align*}$$

  where $x_0 = a\text{,}$ $x_i = a + i\Delta x\text{,}$ and $x_n = b\text{,}$ using $\Delta x = \frac{b-a}{n}\text{.}$ For the midpoint sum, $\overline{x}_{i} = (x_{i-1} + x_i)/2\text{.}$