Sigma Notation and Riemann Sums

Read this section to learn about area. Work through practice problems 1-9.

Sums of Areas of Rectangles

In Section 4.2 we will approximate the areas under curves by building rectangles as high as the curve, calculating the area of each rectangle, and then adding the rectangular areas together.

Example 3: Evaluate the sum of the rectangular areas in Fig. 2, and write the sum using the sigma notation.

Solution:

$\begin{aligned} \{\text { sum of the rectangular areas }\}=&\{\text { sum of (base):(height) for each rectangle }\} \\ &=(1) \cdot(1 / 3)+(1) \cdot(1 / 4)+(1)^{*}(1 / 5)=47 / 60 \end{aligned}$

Using the sigma notation,

$(1) \cdot(1 / 3)+(1) \cdot(1 / 4)+(1) \cdot(1 / 5)=\sum_{k=3}^{5} \frac{1}{k}$

Practice 4: Evaluate the sum of the rectangular areas in Fig. 3, and write the sum using the sigma notation.

The bases of the rectangles do not have to be equal. For the rectangular areas in Fig. 4

rectangle	base	height	area
1	$3-1=2$	$f(2)=4$	$2 \cdot 4=8$
2	$4–3=1$	$f(4)=16$	$1 \cdot 16=16$
3	$6–4=2$	$(f(5)=25$	$2 \cdot 25=50$

so the sum of the rectangular areas is $8 + 16 + 50 = 74$ .

Example 4: Write the sum of the areas of the rectangles in Fig. 5 using the sigma notation.

Solution: The area of each rectangle is $\text { (base) } \cdot \text { (height)}$ .

rectangle	base	height	area
1	$\mathrm{x}_{1}-\mathrm{x}_{0}$	$\mathrm{f}\left(\mathrm{x}_{1}\right)$	$\left(\mathrm{x}_{1}-\mathrm{x}_{0}\right) \cdot \mathrm{f}\left(\mathrm{x}_{1}\right)$
2	$\mathrm{x}_{2}-\mathrm{x}_{1}$	$\mathrm{f}\left(\mathrm{x}_{2}\right)$	$\left(x_{2}-x_{1}\right) \cdot f\left(x_{2}\right)$
3	$\mathrm{x}_{3}-\mathrm{x}_{2}$	$\mathrm{f}\left(\mathrm{x}_{3}\right)$	$\left(x_{3}-x_{2}\right) \cdot f\left(x_{3}\right)$