MA005: Calculus I

Two particular Riemann sums are of special interest because they represent the extreme possibilities for Riemann sums for a given partition.

Definition: Suppose $f$ is a positive function on $[a,b]$, and $P$ is a partition of $[a,b]$. Let $\mathbf{m}_{\mathbf{k}}$ be the x–value in the kth subinterval so that $\mathrm{f}\left(\mathbf{m}_{\mathbf{k}}\right)$ is the minimum value of $f$ in that interval, and let $\mathbf{M}_{\mathbf{k}}$ be the x–value in the kth subinterval so that $\mathrm{f}\left(\mathbf{M}_{\mathbf{k}}\right)$ is the maximum value of $f$ in that interval.

$\mathrm{LS}_{\mathrm{P}}: \sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{f}\left(\mathbf{m}_{\mathbf{k}}\right) \cdot \Delta x_{\mathrm{k}}$ is the lower sum of f for the partition $P$.

$\mathrm{US}_{\mathrm{P}}: \sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{f}\left(\mathbf{M}_{\mathbf{k}}\right) \cdot \Delta x_{\mathrm{k}}$ is the upper sum of f for the partition $P$.

Geometrically, the lower sum comes from building rectangles under the graph of $f$ (Fig. 15a), and the lower sum (every lower sum) is less than or equal to the exact area $\text { A: } L S_{P} \leq A$ for every partition $P$. The upper sum comes from building rectangles over the graph of f (Fig. 15b), and the upper sum (every upper sum) is greater than or equal to the exact area $A$:

$\mathrm{US}_{\mathrm{P}} \geq \mathrm{A}$ for every partition $P$. The lower and upper sums provide bounds on the size of the exact area:

$\mathrm{LS}_{\mathrm{P}} \leq \mathrm{A} \leq \mathrm{US}_{\mathrm{P}}$

For any $\mathrm{c}_{\mathrm{k}}$ value in the kth subinterval, $\mathrm{f}\left(\mathbf{m}_{\mathbf{k}}\right) \leq \mathrm{f}\left(\mathrm{c}_{\mathrm{k}}\right) \leq \mathrm{f}\left(\mathbf{M}_{\mathbf{k}}\right)$, so, for any choice of the $\mathrm{c}_{\mathrm{k}}$ values, the Riemann sum $\mathrm{RS}_{\mathrm{P}}=\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{f}\left(\mathrm{c}_{\mathrm{k}}\right) \cdot \Delta x_{\mathrm{k}}$ satisfies

$\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{f}\left(\mathbf{m}_{\mathbf{k}}\right) \cdot \Delta x_{\mathrm{k}} \leq \sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{f}\left(\mathrm{c}_{\mathrm{k}}\right) \cdot \Delta x_{\mathrm{k}} \leq \sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{f}\left(\mathrm{M}_{\mathbf{k}}\right) \cdot \Delta x_{\mathrm{k}}$ or, equivalently, $\mathrm{LS}_{\mathrm{P}} \leq \mathrm{RS}_{\mathrm{P}} \leq \mathrm{US}_{\mathrm{P}}$.

The lower and upper sums provide bounds on the size of all Riemann sums. The exact area $A$ and every Riemann sum $\mathrm{RS}_{\mathrm{P}}$ for partition $P$ both lie between the lower sum and the upper sum for $P$ (Fig. 16). Therefore, if the lower and upper sums are close together then the area and any Riemann sum for $P$ must also be close together. If we know that the upper and lower sums for a partition $P$ are within 0.001 units of each other, then we can be sure that every Riemann sum for partition $P$ is within 0.001 units of the exact area.

Unfortunately, finding minimums and maximums can be a time–consuming business, and it is usually not practical to determine lower and upper sums for "wiggly" functions. If f is monotonic, however, then it is easy to find the values for $\mathbf{m}_{\mathbf{k}}$ and $\mathbf{M}_{\mathbf{k}}$ , and sometimes we can explicitly calculate the limits of the lower and upper sums.

For a monotonic bounded function we can guarantee that a Riemann sum is within a certain distance of the exact value of the area it is approximating.

Theorem: If $f$ is a positive, montonically increasing, bounded function on $[a,b]$, then for any partition $P$ and any Riemann sum for $P$,

$\left\{\begin{array}{l} \text { distance between the } \\ \text { Riemann sum and } \\ \text { the exact area } \end{array}\right\} \leq\left\{\begin{array}{l} \text { distance between the } \\ \text { upper sum and } \\ \text { the lower sum } \end{array}\right\} \leq\{\mathrm{f}(\mathrm{b})-\mathrm{f}(\mathrm{a})\} \cdot(\text { mesh of } \mathrm{P})$.

Proof: The Riemann sum and the exact area are both between the upper and lower sums so the distance between the Riemann sum and the exact area is less than or equal to the distance between the upper and lower sums. Since $f$ is monotonically increasing, the areas representing the difference of the upper and lower sums can be slid into a rectangle (Fig. 17) whose height equals $f(b)-f(a)$ and whose base equals the mesh of $P$. Then the total difference of the upper and lower sums is less than or equal to the area of the rectangle, $\{\mathrm{f}(\mathrm{b})-f(a)\} \cdot(\text { mesh of } P)$.