[Math] Riemann sums, finding the lower sum

calculusriemann sum

just doing some review and I am a bit confused on how I pick the upper vs lower sum of a Riemann sum.
I get for the upper sum I choose the maximum value of $f$ on $[x_{k-1},x_k]$ and the lower sum is the minimum of $[x_{k-1},x_k]$.

Could someone give me a worked out example of an upper or lower sum using Riemann's definition.

Do I plug in values of the interval $[a,b]$ and see where it is increasing and decreasing? But which formula do I plug this in, the left sum or right sum?

Best Answer

Suppose that $f\colon[-1,1]\longrightarrow\mathbb R$ is defined by $f(x)=x^2$. Now, take $x_0=-1$, $x_1=-\frac13$, $x_2=\frac13$, and $x_3=1$. Then:

  1. in $[x_0,x_1]$, the minimum of $f$ is $\frac19$ and the maximum is $1$;
  2. in $[x_1,x_2]$, the minimum of $f$ is $0$ and the maximum is $\frac19$;
  3. in $[x_2,x_3]$, the minimum of $f$ is $\frac19$ and the maximum is $1$.

Therefore, if $P=\{x_0,x_1,x_2\}$, then the lower sum of $f$ with respect to $P$ is$$\frac23\times\frac19+\frac23\times0+\frac23\times\frac{1}{9}=\frac4{27}$$and the upper sum is$$\frac23\times1+\frac23\times\frac19+\frac23\times\frac19=\frac{38}{27}.$$

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