Summary of Approximating Areas

The use of sigma (summation) notation of the form [latex]\displaystyle\sum_{i=1}^{n}a_i[/latex] is useful for expressing long sums of values in compact form.
For a continuous function defined over an interval [latex][a,b][/latex], the process of dividing the interval into [latex]n[/latex] 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 [latex]\Delta x=\dfrac{b-a}{n}[/latex]
Riemann sums are expressions of the form [latex]\displaystyle\sum_{i=1}^{n} f(x_i^*)\Delta x[/latex], and can be used to estimate the area under the curve [latex]y=f(x)[/latex]. Left- and right-endpoint approximations are special kinds of Riemann sums where the values of [latex]\{x_i^*\}[/latex] 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 [latex]\{x_i^*\}[/latex] 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
[latex]\underset{i=1}{\overset{n}{\Sigma}}c=nc[/latex]
[latex]\underset{i=1}{\overset{n}{\Sigma}}ca_i=c\underset{i=1}{\overset{n}{\Sigma}}a_i[/latex]
[latex]\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[/latex]
[latex]\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[/latex]
[latex]\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[/latex]
Sums and Powers of Integers
[latex]\underset{i=1}{\overset{n}{\Sigma}}i=1+2+\cdots+n=\frac{n(n+1)}{2}[/latex]
[latex]\underset{i=1}{\overset{n}{\Sigma}}i^2=1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}[/latex]
[latex]\underset{i=1}{\overset{n}{\Sigma}}i^3=1^3+2^3+\cdots+n^3=\frac{n^2(n+1)^2}{4}[/latex]
Left-Endpoint Approximation
[latex]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[/latex]
Right-Endpoint Approximation
[latex]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[/latex]

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 [latex]f(x)[/latex] on each subinterval

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 [latex]A\approx \underset{i=1}{\overset{n}{\Sigma}}f(x_i^*)\Delta x[/latex]

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 ([latex]\Sigma[/latex]) 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 [latex]f(x)[/latex] on each subinterval