4.4 Riemann Sums

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We compute Riemann Sums to approximate the area under a curve.

The first major problem in calculus was the tangent line problem. We now turn to the second major problem in calculus, the area problem. We saw that the derivative solved the tangent line problem and it turns out that the anti-derivative solves the area problem.

The Area Problem

Let $f(x)$ be a non-negative, continuous function on the interval $[a,b]$ . The area problem asks us to find the area under the graph of $y = f(x)$ over the interval $[a,b]$ .

To solve this problem, we begin by approximating the area under the curve using rectangles. The sum of the areas of these rectangles is called a Riemann Sum.

To find the exact area under the curve we will need to use infinitely many rectangles. This will lead us into the next section on the Definite Integral. For now, we begin with a special notation that will allow us to write the sum of the areas of many rectangles in a compact form.

Summation Notation

In mathematics, the symbol $\sum$ is used to denote summation.

Summation Notation $\sum _{i = 1}^n \ a_i = a_1 + a_2 + \dots + a_n.$ The parameter $i$ is called the index of summation and it begins at the indicated value, $i=1$ in this case; and it increments by one unit until it reaches the terminal value (in this case, $n$ ).

$\sum _{i = 1}^4 \ (i^2 + 3) = 4 + 7 + 12 + 19 = 42.$

$\sum _{i = 2}^7 \ (2i +5) = 9 + 11 + 13 + 15 + 17 + 19 = 84.$

Find the value of the sum: $\sum _{i = 1}^5 \ (4i -1).$ The sum is $\answer {55}.$

Find the value of the sum: $\sum _{i = 3}^6 \ (i^2 - i + 1).$ The sum is $\answer {72}.$

Special Sums Sums of powers of $i$ :

$\displaystyle {\sum _{i = 1}^n \ i = 1 + 2 + 3+ \cdots + n = \frac {n(n+1)}{2}},$
$\displaystyle {\sum _{i = 1}^n \ i^2 = 1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac {n(n+1)(2n+1)}{6}}$ and
$\displaystyle {\sum _{i = 1}^n \ i^3 = 1^3 + 2^3+ 3^3 + \cdots + n^3 = \frac {n^2(n+1)^2}{4}}$ .

Find the value of the sum: $\sum _{i = 1}^{100} \ i.$ The sum is $\answer {5050}.$

Find the value of the sum: $\sum _{i = 1}^{10} \ i^2.$ The sum is $\answer {385}.$

Find the value of the sum: $\sum _{i = 1}^5 \ i^3.$ The sum is $\answer {225}.$

Riemann Sums

Let $f(x)$ be defined on a closed interval $[a,b]$ . Partition the interval $[a,b]$ into $n$ sub-intervals, each of width $\Delta x = \frac {b-a}{n}.$ Denote the endpoints of the sub-intervals by $x_i$ , so that $x_i = a + i\cdot \Delta x,$ where $i = 0, 1, 2, \ldots , n$ . (Note that $x_0 = a$ and $x_n = b$ .)

To create rectangles under the curve, we next select sample points- one in each sub-interval so as to space them out roughly evenly over the interval $[a,b]$ . The sample points are denoted by an asterisk, like $x_4^*$ for the fourth sample point and $x_i^*$ for the $i$ -th sample point.

In general, the $i$ -th sample point, $x_i^*$ lies in the interval $[x_{i-1}, x_i]$ . The sample points are used to determine the heights of the rectangles used in our Riemann Sum. Specifically, the height of the first rectangle will be $f(x_1^*)$ , the height of the second rectangle will be $f(x_2^*)$ and so on until we reach the $n$ -th and final rectangle, whose height will be $f(x_n^*)$ .

Riemann Sum The Riemann Sum of $f(x)$ on $[a,b]$ with $n$ sub-intervals is given by $\sum _{i = 1}^n \ f(x_i^*) \Delta x,$ where $x_{i-1} \leq x_i^* \leq x_i$ . The numbers $x_i^*$ are called sample points for the sub-intervals $[x_{i-1}, x_i]$ .

For a function $f(x)$ which is positive on the interval $[a,b]$ the Riemann Sum represents the sum of the areas of $n$ rectangles. The heights of the rectangles are given by the function values at the sample points, $f(x_i^*)$ and the widths are given by $\Delta x = (b-a)/n$ .

Below is an interactive graph of the parabola $y=x^2$ . The Riemann Sum uses the rectangles in the figure to approximate the area under the curve. The sample points $x_i^*$ are taken to be endpoints of the sub-interval $[x_{i-1}, x_i]$ . The orange rectangles use $x_i^* = x_{i-1}$ , i.e., a left-endpoint approximation and the purple rectangles use a right-endpoint approximation with $x_i^* = x_i$ . Use the slider to convince yourself that as the number of rectangles, $n$ increases, the width $\Delta x$ decreases and the area of the rectangles approaches the area under the curve.

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Below is the graph corresponding to a Riemann Sum for the curve $f(x) = e^{-x}$ over the interval $[-1,1]$ using 4 rectangles and left endpoints as the sample points.

The value of this Left Riemann Sum is $f(-1) \Delta x + f(-0.5) \Delta x +f(0) \Delta x +f(0.5) \Delta x,$ which equals $e^{1} (0.5) + e^{0.5} (0.5) + e^0 (0.5) + e^{-0.5} (0.5) \approx 2.987.$

Below is the graph corresponding to a Riemann Sum for the curve $f(x) = x^2 + 1$ over the interval $[0,1]$ using 5 rectangles and left endpoints as the sample points.

Compute the Riemann Sum for the function $f(x) = x^2 + 1$ on the interval $[0, 1]$ using $n = 5$ rectangles and choosing the sample points to be left end-points.

The Left Riemann Sum is $\answer {1.24}$ .

Below is the graph corresponding to the Riemann Sum for the curve $f(x) = e^{-x}$ over the interval $[-1,1]$ using 4 rectangles and right endpoints as the sample points.

The value of this Right Riemann Sum is $f(-0.5) \Delta x + f(0) \Delta x +f(0.5) \Delta x + f(1) \Delta x ,$ which equals $e^{0.5} (0.5) + e^0 (0.5) + e^{-0.5} (0.5) + e^{-1} (0.5) \approx 1.812.$

Below is the graph corresponding to a Riemann Sum for the curve $f(x) = x^2 + 1$ over the interval $[0,1]$ using 5 rectangles and right endpoints as the sample points.

Compute the Riemann Sum for the function $f(x) = x^2 + 1$ on the interval $[0, 1]$ using $n = 5$ rectangles and choosing the sample points to be right end-points.

The Right Riemann Sum is $\answer {1.44}$ .