# Riemann sum proof that talks about their graphs [closed]

Questions:

my solutions:

4) $\sum_{i=1}^{n} f(x_i)\Delta x - \sum_{i=1}^{n} f(x_{i-1})\Delta x$ (By definition)

=$\sum_{i=1}^{n} (f(x_i)\Delta x - f(x_{i-1})\Delta x)$ (Sigma notation properties)

= $\sum_{i=1}^{n} (f(x_i) - f(x_{i-1}))\Delta x$ (Factored $\Delta x$)

= $\Delta x (f(b) - f(a))$ (Yeah, I don't know what to say about this.)

= $\frac{b-a}{n} (f(b)-f(a))$ ($\Delta x$ definition) \

5)

(a) If $f(x)$ is positive and increasing, and $f(x) \geq 0, \forall x \in [a,b]$, then

$L_n \leq A \leq R_n$. Let n be any positive integer and let $\Delta x = \frac{b-a}{n}$, such that $x_i = a + i\Delta x$ for i = $0,1,2,\cdots, n$. Since the function is increasing we know that $f(x_{i-1}) \leq f(x_i)$, and also: $$\sum_{i=1}^{n} f(x_{i-1})\Delta x \leq \sum_{i=1}^{n} f(x_{i})$$, $\forall i$ in n rectangles \

(b) If $f(x)$ is positive and decreasing, and $f(x) \geq 0, \forall x \in [a,b]$, then

$R_n \leq A \leq L_n$. Let n be any positive integer and let $\Delta x = \frac{b-a}{n}$, such that $x_i = a + i\Delta x$ for i = $0,1,2,\cdots, n$. Since the function is decreasing we know that $f(x_{i}) \leq f(x_{i-1})$, and also: $$\sum_{i=1}^{n} f(x_{i})\Delta x \leq \sum_{i=1}^{n} f(x_{i-1})$$, $\forall i$ in n rectangles

Are they correct? I also have a question for question 6, does it really matter if it is positive or negative? If the function was negative, then the observation would still be the same, correct?

## closed as unclear what you're asking by Vladhagen, Daniel W. Farlow, JonMark Perry, user91500, user223391 Jan 19 '17 at 22:51

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Yeah, I don't know = Telescoping sum – user305860 Jan 17 '17 at 21:14
• Tinler, I know where these questions are from, I go to same university as you LOL! #The6ix – K Split X Jan 17 '17 at 22:49