The properties associated with the summation process are given in the following rule.
Rule: properties of sigma notation
Let
${a}_{1},{a}_{2}\text{,\u2026,}\phantom{\rule{0.2em}{0ex}}{a}_{n}$ and
${b}_{1},{b}_{2}\text{,\u2026,}\phantom{\rule{0.2em}{0ex}}{b}_{n}$ represent two sequences of terms and let
c be a constant. The following properties hold for all positive integers
n and for integers
m , with
$1\le m\le n.$
A few more formulas for frequently found functions simplify the summation process further. These are shown in the next rule, for
sums and powers of integers , and we use them in the next set of examples.
Now that we have the necessary notation, we return to the problem at hand: approximating the area under a curve. Let
$f\left(x\right)$ be a continuous, nonnegative function defined on the closed interval
$\left[a,b\right].$ We want to approximate the area
A bounded by
$f\left(x\right)$ above, the
x -axis below, the line
$x=a$ on the left, and the line
$x=b$ on the right (
[link] ).
How do we approximate the area under this curve? The approach is a geometric one. By dividing a region into many small shapes that have known area formulas, we can sum these areas and obtain a reasonable estimate of the true area. We begin by dividing the interval
$\left[a,b\right]$ into
n subintervals of equal width,
$\frac{b-a}{n}.$ We do this by selecting equally spaced points
${x}_{0},{x}_{1},{x}_{2}\text{,\u2026,}\phantom{\rule{0.2em}{0ex}}{x}_{n}$ with
${x}_{0}=a,{x}_{n}=b,$ and
${x}_{i}-{x}_{i-1}=\frac{b-a}{n}$
for
$i=1,2,3\text{,\u2026,}\phantom{\rule{0.2em}{0ex}}n.$
We denote the width of each subinterval with the notation Δ
x , so
$\text{\Delta}x=\frac{b-a}{n}$ and
I don't understand the set builder nototation like in this case they've said numbers greater than 1 but less than 5 is there a specific way of reading {x|1<x<5} this because I can't really understand
Yes ☺️ Y=2x+3 will be {2(x+h)+ 3 -(2x+3)}/h where the 2x's and 3's cancel on opening the brackets.
Then from (2h/h)=2
since we have no h for the limit that tends to zero, I guess that is it....
Philip
Correct
Mohamed
thank you Philip Kotia
Rachael
Welcome
Philip
g(x)=8-4x sqrt of 3 + 2x sqrt of 8 what is the answer?
Sheila
I have no idea what these symbols mean can it be explained in English words