Q. 0 - Calculus

Question

Problem Zero: Read the section and make your own sum-

mary of the material.

Step-by-Step Solution

Answer

 Section Riemann Sums state the definition of Riemann Sums, n-rectangle left sum, right sum, midpoint sum, upper sum, and lower sum. 

Step 1. Given information.

The given topic of the section is Riemann Sums.

Step 2. Summary.

If function f is defined on an interval $[a,b]$and $∆\mathrm{x}=\frac{\mathrm{b}-\mathrm{a}}{2}, {\stackrel{*}{\mathrm{x}}}_{\mathrm{k}}=\mathrm{a}+\mathrm{k}∆\mathrm{x}, \& {\stackrel{*}{\mathrm{x}}}_{\mathrm{k}}\in [x_{\mathrm{k}-1},x_{\mathrm{k}}]$ then,

 Riemann Sums for function f is$\sum _{k=1}^{n}f\left({\stackrel{*}{x}}_k\right)∆x$.

the n-rectangle left sum for f is $\sum _{k=1}^{n}f\left({\stackrel{}{x}}_{k-1}\right)∆x.$

the n-rectangle right sum for f is $\sum _{k=1}^{n}f\left({\stackrel{}{x}}_k\right)∆x.$

the n-rectangle midpoint sum for f is $\sum _{k=1}^{n}f\left(\frac{{\stackrel{}{x}}_{k-1}-x_k}{2}\right)∆x.$

the n-rectangle upper sum for f is $\sum _{k=1}^{n}f\left(M_k\right)∆x,$where $f\left(M_k\right)$is the maximum value on $[x_{\mathrm{k}-1},x_{\mathrm{k}}].$

the n-rectangle lower sum for f is $\sum _{k=1}^{n}f\left(m_k\right)∆x,$where $f\left(m_k\right)$is the minimum value on $[x_{\mathrm{k}-1},x_{\mathrm{k}}].$