5.1 Approximating areas  (Page 7/17)

Finding lower and upper sums for $f\left(x\right)=\text{sin}x$

Find a lower sum for $f\left(x\right)=\text{sin}x$ over the interval $\left[a,b\right]=\left[0,\frac{\pi }{2}\right];$ let $n=6.$

Let’s first look at the graph in [link] to get a better idea of the area of interest.

The intervals are $\left[0,\frac{\pi }{12}\right],\left[\frac{\pi }{12},\frac{\pi }{6}\right],\left[\frac{\pi }{6},\frac{\pi }{4}\right],\left[\frac{\pi }{4},\frac{\pi }{3}\right],\left[\frac{\pi }{3},\frac{5\pi }{12}\right],$ and $\left[\frac{5\pi }{12},\frac{\pi }{2}\right].$ Note that $f\left(x\right)=\text{sin}x$ is increasing on the interval $\left[0,\frac{\pi }{2}\right],$ so a left-endpoint approximation gives us the lower sum. A left-endpoint approximation is the Riemann sum ${\displaystyle \sum _{i=0}^5\text{sin}{x}_i\left(\frac{\pi }{12}\right)}.$ We have

$\begin{array}{cc}A\hfill & \approx \text{sin}\left(0\right)\left(\frac{\pi }{12}\right)+\text{sin}\left(\frac{\pi }{12}\right)\left(\frac{\pi }{12}\right)+\text{sin}\left(\frac{\pi }{6}\right)\left(\frac{\pi }{12}\right)+\text{sin}\left(\frac{\pi }{4}\right)\left(\frac{\pi }{12}\right)+\text{sin}\left(\frac{\pi }{3}\right)\left(\frac{\pi }{12}\right)+\text{sin}\left(\frac{5\pi }{12}\right)\left(\frac{\pi }{12}\right)\hfill \\ & =\mathrm{0.863.}\hfill \end{array}$

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Key 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 $\left[a,b\right],$ 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 $\text{Δ}x=\frac{b-a}{n}.$
• Riemann sums are expressions of the form ${\displaystyle \sum _{i=1}^{n}f\left(x_i^{*}\right)\text{Δ}x},$ and can be used to estimate the area under the curve $y=f\left(x\right).$ Left- and right-endpoint approximations are special kinds of Riemann sums where the values of $\left\{x_i^{*}\right\}$ 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 $\left\{x_i^{*}\right\}$ 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
  ${\displaystyle \sum _{i=1}^{n}c}=nc$
  ${\displaystyle \sum _{i=1}^{n}c{a}_i}=c{\displaystyle \sum _{i=1}^n{a}_i}$
  ${\displaystyle \sum _{i=1}^n\left(a_i+b_i\right)}={\displaystyle \sum _{i=1}^n{a}_i}+{\displaystyle \sum _{i=1}^n{b}_i}$
  ${\displaystyle \sum _{i=1}^n\left(a_i-b_i\right)}={\displaystyle \sum _{i=1}^n{a}_i}-{\displaystyle \sum _{i=1}^n{b}_i}$
  ${\displaystyle \sum _{i=1}^n{a}_i}={\displaystyle \sum _{i=1}^m{a}_i}+{\displaystyle \sum _{i=m+1}^n{a}_i}$
• Sums and Powers of Integers
  ${\displaystyle \sum _{i=1}^{n}i}=1+2+\text{⋯}+n=\frac{n\left(n+1\right)}{2}$
  ${\displaystyle \sum _{i=1}^n{i}^2}=1^2+2^2+\text{⋯}+n^2=\frac{n\left(n+1\right)\left(2n+1\right)}{6}$
  ${\displaystyle \sum _{i=0}^n{i}^3}=1^3+2^3+\text{⋯}+n^3=\frac{n^2{\left(n+1\right)}^2}{4}$
• Left-Endpoint Approximation
  $A\approx L_n=f\left(x_0\right)\text{Δ}x+f\left(x_1\right)\text{Δ}x+\text{⋯}+f\left(x_{n-1}\right)\text{Δ}x={\displaystyle \sum _{i=1}^{n}f\left(x_{i-1}\right)\text{Δ}x}$
• Right-Endpoint Approximation
  $A\approx R_n=f\left(x_1\right)\text{Δ}x+f\left(x_2\right)\text{Δ}x+\text{⋯}+f\left(x_n\right)\text{Δ}x={\displaystyle \sum _{i=1}^{n}f\left(x_i\right)\text{Δ}x}$

In the following exercises, use the rules for sums of powers of integers to compute the sums.

Suppose that ${\displaystyle \sum _{i=1}^{100}{a}_i=15}$ and ${\displaystyle \sum _{i=1}^{100}{b}_i=\mathrm{-12}}.$ In the following exercises, compute the sums.

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.