## Summary of Approximating Areas

### Essential Concepts

• The use of sigma (summation) notation of the form $\displaystyle\sum_{i=1}^{n}a_i$ is useful for expressing long sums of values in compact form.
• For a continuous function defined over an interval $[a,b]$, the process of dividing the interval into $n$ equal parts, extending a rectangle to the graph of the function, calculating the areas of the series of rectangles, and then summing the areas yields an approximation of the area of that region.
• The width of each rectangle is $\Delta x=\dfrac{b-a}{n}$
• Riemann sums are expressions of the form $\displaystyle\sum_{i=1}^{n} f(x_i^*)\Delta x$, and can be used to estimate the area under the curve $y=f(x)$. Left- and right-endpoint approximations are special kinds of Riemann sums where the values of $\{x_i^*\}$ are chosen to be the left or right endpoints of the subintervals, respectively.
• Riemann sums allow for much flexibility in choosing the set of points $\{x_i^*\}$ at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum.

## Key Equations

• Properties of Sigma Notation
$\underset{i=1}{\overset{n}{\Sigma}}c=nc$
$\underset{i=1}{\overset{n}{\Sigma}}ca_i=c\underset{i=1}{\overset{n}{\Sigma}}a_i$
$\underset{i=1}{\overset{n}{\Sigma}}(a_i+b_i)=\underset{i=1}{\overset{n}{\Sigma}}a_i+\underset{i=1}{\overset{n}{\Sigma}}b_i$
$\underset{i=1}{\overset{n}{\Sigma}}(a_i-b_i)=\underset{i=1}{\overset{n}{\Sigma}}a_i-\underset{i=1}{\overset{n}{\Sigma}}b_i$
$\underset{i=1}{\overset{n}{\Sigma}}a_i=\underset{i=1}{\overset{m}{\Sigma}}a_i+\underset{i=m+1}{\overset{n}{\Sigma}}a_i$
• Sums and Powers of Integers
$\underset{i=1}{\overset{n}{\Sigma}}i=1+2+\cdots+n=\frac{n(n+1)}{2}$
$\underset{i=1}{\overset{n}{\Sigma}}i^2=1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$
$\underset{i=1}{\overset{n}{\Sigma}}i^3=1^3+2^3+\cdots+n^3=\frac{n^2(n+1)^2}{4}$
• Left-Endpoint Approximation
$A \approx L_n=f(x_0)\Delta x+f(x_1)\Delta x+\cdots+f(x_{n-1})\Delta x=\underset{i=1}{\overset{n}{\Sigma}}f(x_{i-1})\Delta x$
• Right-Endpoint Approximation
$A \approx R_n=f(x_1)\Delta x+f(x_2)\Delta x+\cdots+f(x_n)\Delta x=\underset{i=1}{\overset{n}{\Sigma}}f(x_i)\Delta x$

## Glossary

left-endpoint approximation
an approximation of the area under a curve computed by using the left endpoint of each subinterval to calculate the height of the vertical sides of each rectangle
lower sum
a sum obtained by using the minimum value of $f(x)$ on each subinterval
partition
a set of points that divides an interval into subintervals
regular partition
a partition in which the subintervals all have the same width
riemann sum
an estimate of the area under the curve of the form $A\approx \underset{i=1}{\overset{n}{\Sigma}}f(x_i^*)\Delta x$
right-endpoint approximation
the right-endpoint approximation is an approximation of the area of the rectangles under a curve using the right endpoint of each subinterval to construct the vertical sides of each rectangle
sigma notation
(also, summation notation) the Greek letter sigma ($\Sigma$) indicates addition of the values; the values of the index above and below the sigma indicate where to begin the summation and where to end it
upper sum
a sum obtained by using the maximum value of $f(x)$ on each subinterval