# 5.1 Approximating areas  (Page 2/17)

 Page 2 / 17

Write in sigma notation and evaluate the sum of terms 2 i for $i=3,4,5,6.$

$\sum _{i=3}^{6}{2}^{i}={2}^{3}+{2}^{4}+{2}^{5}+{2}^{6}=120$

The properties associated with the summation process are given in the following rule.

## Rule: properties of sigma notation

Let ${a}_{1},{a}_{2}\text{,…,}\phantom{\rule{0.2em}{0ex}}{a}_{n}$ and ${b}_{1},{b}_{2}\text{,…,}\phantom{\rule{0.2em}{0ex}}{b}_{n}$ 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 $1\le m\le n.$

1. $\sum _{i=1}^{n}c=nc$

2. $\sum _{i=1}^{n}c{a}_{i}=c\sum _{i=1}^{n}{a}_{i}$

3. $\sum _{i=1}^{n}\left({a}_{i}+{b}_{i}\right)=\sum _{i=1}^{n}{a}_{i}+\sum _{i=1}^{n}{b}_{i}$

4. $\sum _{i=1}^{n}\left({a}_{i}-{b}_{i}\right)=\sum _{i=1}^{n}{a}_{i}-\sum _{i=1}^{n}{b}_{i}$

5. $\sum _{i=1}^{n}{a}_{i}=\sum _{i=1}^{m}{a}_{i}+\sum _{i=m+1}^{n}{a}_{i}$

## Proof

We prove properties 2. and 3. here, and leave proof of the other properties to the Exercises.

2. We have

$\begin{array}{cc}\sum _{i=1}^{n}c{a}_{i}\hfill & =c{a}_{1}+c{a}_{2}+c{a}_{3}+\text{⋯}+c{a}_{n}\hfill \\ & =c\left({a}_{1}+{a}_{2}+{a}_{3}+\text{⋯}+{a}_{n}\right)\hfill \\ \\ \\ & =c\sum _{i=1}^{n}{a}_{i}.\hfill \end{array}$

3. We have

$\begin{array}{cc}\sum _{i=1}^{n}\left({a}_{i}+{b}_{i}\right)\hfill & =\left({a}_{1}+{b}_{1}\right)+\left({a}_{2}+{b}_{2}\right)+\left({a}_{3}+{b}_{3}\right)+\text{⋯}+\left({a}_{n}+{b}_{n}\right)\hfill \\ & =\left({a}_{1}+{a}_{2}+{a}_{3}+\text{⋯}+{a}_{n}\right)+\left({b}_{1}+{b}_{2}+{b}_{3}+\text{⋯}+{b}_{n}\right)\hfill \\ \\ \\ & =\sum _{i=1}^{n}{a}_{i}+\sum _{i=1}^{n}{b}_{i}.\hfill \end{array}$

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.

## Rule: sums and powers of integers

1. The sum of n integers is given by
$\sum _{i=1}^{n}i=1+2+\text{⋯}+n=\frac{n\left(n+1\right)}{2}.$
2. The sum of consecutive integers squared is given by
$\sum _{i=1}^{n}{i}^{2}={1}^{2}+{2}^{2}+\text{⋯}+{n}^{2}=\frac{n\left(n+1\right)\left(2n+1\right)}{6}.$
3. The sum of consecutive integers cubed is given by
$\sum _{i=1}^{n}{i}^{3}={1}^{3}+{2}^{3}+\text{⋯}+{n}^{3}=\frac{{n}^{2}{\left(n+1\right)}^{2}}{4}.$

## Evaluation using sigma notation

Write using sigma notation and evaluate:

1. The sum of the terms ${\left(i-3\right)}^{2}$ for $i=1,2\text{,…,}\phantom{\rule{0.2em}{0ex}}200.$
2. The sum of the terms $\left({i}^{3}-{i}^{2}\right)$ for $i=1,2,3,4,5,6.$
1. Multiplying out ${\left(i-3\right)}^{2},$ we can break the expression into three terms.
$\begin{array}{cc}\sum _{i=1}^{200}{\left(i-3\right)}^{2}\hfill & =\sum _{i=1}^{200}\left({i}^{2}-6i+9\right)\hfill \\ \\ \\ & =\sum _{i=1}^{200}{i}^{2}-\sum _{i=1}^{200}6i+\sum _{i=1}^{200}9\hfill \\ & =\sum _{i=1}^{200}{i}^{2}-6\sum _{i=1}^{200}i+\sum _{i=1}^{200}9\hfill \\ & =\frac{200\left(200+1\right)\left(400+1\right)}{6}-6\left[\frac{200\left(200+1\right)}{2}\right]+9\left(200\right)\hfill \\ & =2,686,700-120,600+1800\hfill \\ & =2,567,900\hfill \end{array}$
2. Use sigma notation property iv. and the rules for the sum of squared terms and the sum of cubed terms.
$\begin{array}{cc}\sum _{i=1}^{6}\left({i}^{3}-{i}^{2}\right)\hfill & =\sum _{i=1}^{6}{i}^{3}-\sum _{i=1}^{6}{i}^{2}\hfill \\ \\ \\ \\ & =\frac{{6}^{2}{\left(6+1\right)}^{2}}{4}-\frac{6\left(6+1\right)\left(2\left(6\right)+1\right)}{6}\hfill \\ & =\frac{1764}{4}-\frac{546}{6}\hfill \\ & =350\hfill \end{array}$

Find the sum of the values of $4+3i$ for $i=1,2\text{,…,}\phantom{\rule{0.2em}{0ex}}100.$

15,550

## Finding the sum of the function values

Find the sum of the values of $f\left(x\right)={x}^{3}$ over the integers $1,2,3\text{,…,}\phantom{\rule{0.2em}{0ex}}10.$

Using the formula, we have

$\begin{array}{cc}\sum _{i=0}^{10}{i}^{3}\hfill & =\frac{{\left(10\right)}^{2}{\left(10+1\right)}^{2}}{4}\hfill \\ \\ & =\frac{100\left(121\right)}{4}\hfill \\ & =3025.\hfill \end{array}$

Evaluate the sum indicated by the notation $\sum _{k=1}^{20}\left(2k+1\right).$

440

## Approximating area

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 ${x}_{0},{x}_{1},{x}_{2}\text{,…,}\phantom{\rule{0.2em}{0ex}}{x}_{n}$ with ${x}_{0}=a,{x}_{n}=b,$ and

${x}_{i}-{x}_{i-1}=\frac{b-a}{n}$

for $i=1,2,3\text{,…,}\phantom{\rule{0.2em}{0ex}}n.$

We denote the width of each subinterval with the notation Δ x , so $\text{Δ}x=\frac{b-a}{n}$ and

${x}_{i}={x}_{0}+i\text{Δ}x$

Differentiation and integration
yes
Damien
proper definition of derivative
the maximum rate of change of one variable with respect to another variable
terms of an AP is 1/v and the vth term is 1/u show that the sum of uv terms is 1/2(uv+1)
what is calculus?
calculus is math that studies the change in math, such as the rate and distance,
Tamarcus
what are the topics in calculus
Augustine
what is limit of a function?
what is x and how x=9.1 take?
what is f(x)
the function at x
Marc
also known as the y value so I could say y=2x or f(x)= 2x same thing just using functional notation your next question is what is dependent and independent variables. I am Dyslexic but know math and which is which confuses me. but one can vary the x value while y depends on which x you use. also
Marc
up domain and range
Marc
enjoy your work and good luck
Marc
I actually wanted to ask another questions on sets if u dont mind please?
Inembo
I have so many questions on set and I really love dis app I never believed u would reply
Inembo
Hmm go ahead and ask you got me curious too much conversation here
am sorry for disturbing I really want to know math that's why *I want to know the meaning of those symbols in sets* e.g n,U,A', etc pls I want to know it and how to solve its problems
Inembo
and how can i solve a question like dis *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
next questions what do dy mean by (A' n B^c)^c'
Inembo
The sets help you to define the function. The function is like a magic box where you put inside stuff(numbers or sets) and you get out the stuff but in different shapes (forms).
I dont understand what you wanna say by (A' n B^c)^c'
(A' n B (rise to the power of c)) all rise to the power of c
Inembo
Aaaahh
Ok so the set is formed by vectors and not numbers
A vector of length n
But you can make a set out of matrixes as well
I I don't even understand sets I wat to know d meaning of all d symbolsnon sets
Inembo
High-school?
yes
Inembo
am having big problem understanding sets more than other math topics
Inembo
So f:R->R means that the function takes real numbers and provides real numer. For ex. If f(x) =2x this means if you give to your function a real number like 2,it gives you also a real number 2times2=4
pls answer this question *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
If you have f:R^n->R^n you give to your function a vector of length n like (a1,a2,...an) where all a1,.. an are reals and gives you also a vector of length n... I don't know if i answering your question. Otherwise on YouTube you havr many videos where they explain it in a simple way
I would say 24
Offer both
Sorry 20
Actually you have 40 - 4 =36 who offer maths or physics or both.
I know its 20 but how to prove it
Inembo
You have 32+24=56who offer courses
56-36=20 who give both courses... I would say that
solution: In a question involving sets and Venn diagram, the sum of the members of set A + set B - the joint members of both set A and B + the members that are not in sets A or B = the total members of the set. In symbolic form n(A U B) = n(A) + n (B) - n (A and B) + n (A U B)'.
Mckenzie
In the case of sets A and B use the letters m and p to represent the sets and we have: n (M U P) = 40; n (M) = 24; n (P) = 32; n (M and P) = unknown; n (M U P)' = 4
Mckenzie
Now substitute the numerical values for the symbolic representation 40 = 24 + 32 - n(M and P) + 4 Now solve for the unknown using algebra: 40 = 24 + 32+ 4 - n(M and P) 40 = 60 - n(M and P) Add n(M and P), as well, subtract 40 from both sides of the equation to find the answer.
Mckenzie
40 - 40 + n(M and P) = 60 - 40 - n(M and P) + n(M and P) Solution: n(M and P) = 20
Mckenzie
thanks
Inembo
Simpler form: Add the sums of set M, set P and the complement of the union of sets M and P then subtract the number of students from the total.
Mckenzie
n(M and P) = (32 + 24 + 4) - 40 = 60 - 40 = 20
Mckenzie
how do i evaluate integral of x^1/2 In x
first you simplify the given expression, which gives (x^2/2). Then you now integrate the above simplified expression which finally gives( lnx^2).
by using integration product formula
Roha
find derivative f(x)=1/x
-1/x^2, use the chain rule
Andrew
f(x)=x^3-2x
Mul
what is domin in this question
noman
all real numbers . except zero
Roha
please try to guide me how?
Meher
what do u want to ask
Roha
?
Roha
the domain of the function is all real number excluding zero, because the rational function 1/x is a representation of a fractional equation (precisely inverse function). As in elementary mathematics the concept of dividing by zero is nonexistence, so zero will not make the fractional statement
Mckenzie
a function's answer/range should not be in the form of 1/0 and there should be no imaginary no. say square root of any negative no. (-1)^1/2
Roha
domain means everywhere along the x axis. since this function is not discontinuous anywhere along the x axis, then the domain is said to be all values of x.
Andrew
Derivative of a function
Waqar
right andrew ... this function is only discontinuous at 0
Roha
of sorry, I didn't realize he was taking about the function 1/x ...I thought he was referring to the function x^3-2x.
Andrew
yep...it's 1/x...!!!
Roha
true and cannot be apart of the domain that makes up the relation of the graph y = 1/x. The value of the denominator of the rational function can never be zero, because the result of the output value (range value of the graph when x =0) is undefined.
Mckenzie
👍
Roha
Therefore, when x = 0 the image of the rational function does not exist at this domain value, but exist at all other x values (domain) that makes the equation functional, and the graph drawable.
Mckenzie
👍
Roha
Roha are u A Student
Lutf
yes
Roha
What is the first fundermental theory of Calculus?
do u mean fundamental theorem ?
Roha
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Mubarak
What is a independent variable
a variable that does not depend on another.
Andrew
which can be any no... does not need to find its value by any other variable.. often x is independent and y is dependent
Roha
solve number one step by step
x-xcosx/sinsq.3x
Hasnain
x-xcosx/sin^23x
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how to prove 1-sinx/cos x= cos x/-1+sin x?
1-sin x/cos x= cos x/-1+sin x
Rochel
how to prove 1-sun x/cos x= cos x / -1+sin x?
Rochel