Summary of Approximating Areas

Essential Concepts

  • 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]

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 [latex]f(x)[/latex] 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 [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