Riemann sums

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We learn the definition of the Definite Integral and we compute Riemann Sums.

In mathematics, the symbol $\sum$ is often 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 $\answer {72}.$

Here are some well know sums:

$\begin{align*} \sum_{i = 1}^n \ i &= \frac{n(n+1)}{2},\\ \sum_{i = 1}^n \ i^2 &= \frac{n(n+1)(2n+1)}{6},\\ \sum_{i = 1}^n \ i^3 &= \frac{n^2(n+1)^2}{4}. \end{align*}$

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 $[a,b]$ . Subdivide 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 $x_n = b$ .)

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 applet that shows Riemann Sums visually. You can select the number of rectangles, $n$ and the location of the sample points $x_i^*$ within the sub-intervals.

Left and Right Riemann Sums

Typically, the sample points $x_i^*$ are chosen according to a prescribed rule, such as ‘‘ $x_i^* =$ left end point’’, i.e. $x_i^* = x_{i-1}$ , or ‘‘ $x_i^* =$ right end point’’, i.e. $x_i^* = x_i$ .
Below is an applet that shows Riemann Sums using left and right endpoints as the sample points.

Upper and Lower Sums

We can also choose the sample points $x_i^*$ to be at locations where the function $f(x)$ has a certain property. For example, we can choose $x_i^*$ such that $f(x)$ has an absolute max or min on the interval $[x_{i-1}, x_i]$ at $x = x_i^*$ . The Riemann sums in these cases are called upper or lower sums, respectively.
Below is an applet where the sample points are chosen so that the Riemann Sums are upper and lower sums.

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 a 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 $\answer {1.44}$ .