# 5.1 Approximating areas  (Page 7/17)

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1. Find an upper sum for $f\left(x\right)=10-{x}^{2}$ on $\left[1,2\right];$ let $n=4.$
2. Sketch the approximation.
1. $\text{Upper sum}=8.0313.$

## Finding lower and upper sums for $f\left(x\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x$

Find a lower sum for $f\left(x\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x$ over the interval $\left[a,b\right]=\left[0,\frac{\pi }{2}\right];$ let $n=6.$

Let’s first look at the graph in [link] to get a better idea of the area of interest.

The intervals are $\left[0,\frac{\pi }{12}\right],\left[\frac{\pi }{12},\frac{\pi }{6}\right],\left[\frac{\pi }{6},\frac{\pi }{4}\right],\left[\frac{\pi }{4},\frac{\pi }{3}\right],\left[\frac{\pi }{3},\frac{5\pi }{12}\right],$ and $\left[\frac{5\pi }{12},\frac{\pi }{2}\right].$ Note that $f\left(x\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x$ is increasing on the interval $\left[0,\frac{\pi }{2}\right],$ so a left-endpoint approximation gives us the lower sum. A left-endpoint approximation is the Riemann sum $\sum _{i=0}^{5}\text{sin}\phantom{\rule{0.1em}{0ex}}{x}_{i}\left(\frac{\pi }{12}\right).$ We have

$\begin{array}{cc}A\hfill & \approx \text{sin}\left(0\right)\left(\frac{\pi }{12}\right)+\text{sin}\left(\frac{\pi }{12}\right)\left(\frac{\pi }{12}\right)+\text{sin}\left(\frac{\pi }{6}\right)\left(\frac{\pi }{12}\right)+\text{sin}\left(\frac{\pi }{4}\right)\left(\frac{\pi }{12}\right)+\text{sin}\left(\frac{\pi }{3}\right)\left(\frac{\pi }{12}\right)+\text{sin}\left(\frac{5\pi }{12}\right)\left(\frac{\pi }{12}\right)\hfill \\ & =0.863.\hfill \end{array}$

Using the function $f\left(x\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x$ over the interval $\left[0,\frac{\pi }{2}\right],$ find an upper sum; let $n=6.$

$A\approx 1.125$

## Key concepts

• The use of sigma (summation) notation of the form $\sum _{i=1}^{n}{a}_{i}$ is useful for expressing long sums of values in compact form.
• For a continuous function defined over an interval $\left[a,b\right],$ the process of dividing the interval into n equal parts, extending a rectangle to the graph of the function, calculating the areas of the series of rectangles, and then summing the areas yields an approximation of the area of that region.
• The width of each rectangle is $\text{Δ}x=\frac{b-a}{n}.$
• Riemann sums are expressions of the form $\sum _{i=1}^{n}f\left({x}_{i}^{*}\right)\text{Δ}x,$ and can be used to estimate the area under the curve $y=f\left(x\right).$ Left- and right-endpoint approximations are special kinds of Riemann sums where the values of $\left\{{x}_{i}^{*}\right\}$ are chosen to be the left or right endpoints of the subintervals, respectively.
• Riemann sums allow for much flexibility in choosing the set of points $\left\{{x}_{i}^{*}\right\}$ at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum.

## Key equations

• Properties of Sigma Notation
$\sum _{i=1}^{n}c=nc$
$\sum _{i=1}^{n}c{a}_{i}=c\sum _{i=1}^{n}{a}_{i}$
$\sum _{i=1}^{n}\left({a}_{i}+{b}_{i}\right)=\sum _{i=1}^{n}{a}_{i}+\sum _{i=1}^{n}{b}_{i}$
$\sum _{i=1}^{n}\left({a}_{i}-{b}_{i}\right)=\sum _{i=1}^{n}{a}_{i}-\sum _{i=1}^{n}{b}_{i}$
$\sum _{i=1}^{n}{a}_{i}=\sum _{i=1}^{m}{a}_{i}+\sum _{i=m+1}^{n}{a}_{i}$
• Sums and Powers of Integers
$\sum _{i=1}^{n}i=1+2+\text{⋯}+n=\frac{n\left(n+1\right)}{2}$
$\sum _{i=1}^{n}{i}^{2}={1}^{2}+{2}^{2}+\text{⋯}+{n}^{2}=\frac{n\left(n+1\right)\left(2n+1\right)}{6}$
$\sum _{i=0}^{n}{i}^{3}={1}^{3}+{2}^{3}+\text{⋯}+{n}^{3}=\frac{{n}^{2}{\left(n+1\right)}^{2}}{4}$
• Left-Endpoint Approximation
$A\approx {L}_{n}=f\left({x}_{0}\right)\text{Δ}x+f\left({x}_{1}\right)\text{Δ}x+\text{⋯}+f\left({x}_{n-1}\right)\text{Δ}x=\sum _{i=1}^{n}f\left({x}_{i-1}\right)\text{Δ}x$
• Right-Endpoint Approximation
$A\approx {R}_{n}=f\left({x}_{1}\right)\text{Δ}x+f\left({x}_{2}\right)\text{Δ}x+\text{⋯}+f\left({x}_{n}\right)\text{Δ}x=\sum _{i=1}^{n}f\left({x}_{i}\right)\text{Δ}x$

State whether the given sums are equal or unequal.

1. $\sum _{i=1}^{10}i$ and $\sum _{k=1}^{10}k$
2. $\sum _{i=1}^{10}i$ and $\sum _{i=6}^{15}\left(i-5\right)$
3. $\sum _{i=1}^{10}i\left(i-1\right)$ and $\sum _{j=0}^{9}\left(j+1\right)j$
4. $\sum _{i=1}^{10}i\left(i-1\right)$ and $\sum _{k=1}^{10}\left({k}^{2}-k\right)$

a. They are equal; both represent the sum of the first 10 whole numbers. b. They are equal; both represent the sum of the first 10 whole numbers. c. They are equal by substituting $j=i-1.$ d. They are equal; the first sum factors the terms of the second.

In the following exercises, use the rules for sums of powers of integers to compute the sums.

$\sum _{i=5}^{10}i$

$\sum _{i=5}^{10}{i}^{2}$

$385-30=355$

Suppose that $\sum _{i=1}^{100}{a}_{i}=15$ and $\sum _{i=1}^{100}{b}_{i}=-12.$ In the following exercises, compute the sums.

$\sum _{i=1}^{100}\left({a}_{i}+{b}_{i}\right)$

$\sum _{i=1}^{100}\left({a}_{i}-{b}_{i}\right)$

$15-\left(-12\right)=27$

$\sum _{i=1}^{100}\left(3{a}_{i}-4{b}_{i}\right)$

$\sum _{i=1}^{100}\left(5{a}_{i}+4{b}_{i}\right)$

$5\left(15\right)+4\left(-12\right)=27$

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.

$\sum _{k=1}^{20}100\left({k}^{2}-5k+1\right)$

$\sum _{j=1}^{50}\left({j}^{2}-2j\right)$

$\sum _{j=1}^{50}{j}^{2}-2\sum _{j=1}^{50}j=\frac{\left(50\right)\left(51\right)\left(101\right)}{6}-\frac{2\left(50\right)\left(51\right)}{2}=40,\text{​}375$

$\sum _{j=11}^{20}\left({j}^{2}-10j\right)$

$\sum _{k=1}^{25}\left[{\left(2k\right)}^{2}-100k\right]$

$4\sum _{k=1}^{25}{k}^{2}-100\sum _{k=1}^{25}k=\frac{4\left(25\right)\left(26\right)\left(51\right)}{9}-50\left(25\right)\left(26\right)=-10,\text{​}400$

questions solve y=sin x
Solve it for what?
Tim
you have to apply the function arcsin in both sides and you get arcsin y = acrsin (sin x) the the function arcsin and function sin cancel each other so the ecuation becomes arcsin y = x you can also write x= arcsin y
Ioana
what is the question ? what is the answer?
Suman
there is an equation that should be solve for x
Ioana
ok solve it
Suman
are you saying y is of sin(x) y=sin(x)/sin of both sides to solve for x... therefore y/sin =x
Tyron
or solve for sin(x) via the unit circle
Tyron
what is unit circle
Suman
a circle whose radius is 1.
Darnell
the unit circle is covered in pre cal...and or trigonometry. it is the multipcation table of upper level mathematics.
Tyron
what is function?
A set of points in which every x value (domain) corresponds to exactly one y value (range)
Tim
what is lim (x,y)~(0,0) (x/y)
limited of x,y at 0,0 is nt defined
Alswell
But using L'Hopitals rule is x=1 is defined
Alswell
Could U explain better boss?
emmanuel
value of (x/y) as (x,y) tends to (0,0) also whats the value of (x+y)/(x^2+y^2) as (x,y) tends to (0,0)
NIKI
can we apply l hospitals rule for function of two variables
NIKI
why n does not equal -1
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
proof the formula integration of udv=uv-integration of vdu.?
Find derivative (2x^3+6xy-4y^2)^2
no x=2 is not a function, as there is nothing that's changing.
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
y=800
800
Bg
how do u factor the numerator?
Nonsense, you factor numbers
Antonio
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
Antonio
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
2x^3+6xy-4y^2)^2 solve this
femi
moe
volume between cone z=√(x^2+y^2) and plane z=2
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
How do we find the horizontal asymptote of a function using limits?
Easy lim f(x) x-->~ =c
Antonio
solutions for combining functions
what is a function? f(x)
one that is one to one, one that passes the vertical line test
Andrew
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
Antonio
is x=2 a function?
The
restate the problem. and I will look. ty
is x=2 a function?
The