# 6.5 Physical applications  (Page 2/11)

 Page 2 / 11

We now approximate the density and area of the washer to calculate an approximate mass, ${m}_{i}.$ Note that the area of the washer is given by

$\begin{array}{cc}\hfill {A}_{i}& =\pi {\left({x}_{i}\right)}^{2}-\pi {\left({x}_{i-1}\right)}^{2}\hfill \\ & =\pi \left[{x}_{i}^{2}-{x}_{i-1}^{2}\right]\hfill \\ & =\pi \left({x}_{i}+{x}_{i-1}\right)\left({x}_{i}-{x}_{i-1}\right)\hfill \\ & =\pi \left({x}_{i}+{x}_{i-1}\right)\text{Δ}x.\hfill \end{array}$

You may recall that we had an expression similar to this when we were computing volumes by shells. As we did there, we use ${x}_{i}^{*}\approx \left({x}_{i}+{x}_{i-1}\right)\text{/}2$ to approximate the average radius of the washer. We obtain

${A}_{i}=\pi \left({x}_{i}+{x}_{i-1}\right)\text{Δ}x\approx 2\pi {x}_{i}^{*}\text{Δ}x.$

Using $\rho \left({x}_{i}^{*}\right)$ to approximate the density of the washer, we approximate the mass of the washer by

${m}_{i}\approx 2\pi {x}_{i}^{*}\rho \left({x}_{i}^{*}\right)\text{Δ}x.$

Adding up the masses of the washers, we see the mass $m$ of the entire disk is approximated by

$m=\sum _{i=1}^{n}{m}_{i}\approx \sum _{i=1}^{n}2\pi {x}_{i}^{*}\rho \left({x}_{i}^{*}\right)\text{Δ}x.$

We again recognize this as a Riemann sum, and take the limit as $n\to \infty .$ This gives us

$m=\underset{n\to \infty }{\text{lim}}\sum _{i=1}^{n}2\pi {x}_{i}^{*}\rho \left({x}_{i}^{*}\right)\text{Δ}x={\int }_{0}^{r}2\pi x\rho \left(x\right)dx.$

We summarize these findings in the following theorem.

## Mass–density formula of a circular object

Let $\rho \left(x\right)$ be an integrable function representing the radial density of a disk of radius $r.$ Then the mass of the disk is given by

$m={\int }_{0}^{r}2\pi x\rho \left(x\right)dx.$

## Calculating mass from radial density

Let $\rho \left(x\right)=\sqrt{x}$ represent the radial density of a disk. Calculate the mass of a disk of radius 4.

Applying the formula, we find

$\begin{array}{cc}\hfill m& ={\int }_{0}^{r}2\pi x\rho \left(x\right)dx\hfill \\ & ={\int }_{0}^{4}2\pi x\sqrt{x}dx=2\pi {\int }_{0}^{4}{x}^{3\text{/}2}dx\hfill \\ & =2\pi {\frac{2}{5}{x}^{5\text{/}2}|}_{0}^{4}=\frac{4\pi }{5}\left[32\right]=\frac{128\pi }{5}.\hfill \end{array}$

Let $\rho \left(x\right)=3x+2$ represent the radial density of a disk. Calculate the mass of a disk of radius 2.

$24\pi$

## Work done by a force

We now consider work. In physics, work is related to force, which is often intuitively defined as a push or pull on an object. When a force moves an object, we say the force does work on the object. In other words, work can be thought of as the amount of energy it takes to move an object. According to physics, when we have a constant force, work can be expressed as the product of force and distance.

In the English system, the unit of force is the pound and the unit of distance is the foot, so work is given in foot-pounds. In the metric system, kilograms and meters are used. One newton is the force needed to accelerate $1$ kilogram of mass at the rate of $1$ m/sec 2 . Thus, the most common unit of work is the newton-meter. This same unit is also called the joule . Both are defined as kilograms times meters squared over seconds squared $\left(\text{kg}·{\text{m}}^{2}\text{/}{\text{s}}^{2}\right).$

When we have a constant force, things are pretty easy. It is rare, however, for a force to be constant. The work done to compress (or elongate) a spring, for example, varies depending on how far the spring has already been compressed (or stretched). We look at springs in more detail later in this section.

Suppose we have a variable force $F\left(x\right)$ that moves an object in a positive direction along the x -axis from point $a$ to point $b.$ To calculate the work done, we partition the interval $\left[a,b\right]$ and estimate the work done over each subinterval. So, for $i=0,1,2\text{,…},n,$ let $P=\left\{{x}_{i}\right\}$ be a regular partition of the interval $\left[a,b\right],$ and for $i=1,2\text{,…},n,$ choose an arbitrary point ${x}_{i}^{*}\in \left[{x}_{i-1},{x}_{i}\right].$ To calculate the work done to move an object from point ${x}_{i-1}$ to point ${x}_{i},$ we assume the force is roughly constant over the interval, and use $F\left({x}_{i}^{*}\right)$ to approximate the force. The work done over the interval $\left[{x}_{i-1},{x}_{i}\right],$ then, is given by

#### Questions & Answers

What is the derivative of 3x to the negative 3
-3x^-4
Mugen
that's wrong, its -9x^-3
Mugen
Or rather kind of the combination of the two: -9x^(-4) I think. :)
Csaba
-9x-4 (X raised power negative 4=
Simon
-9x-³
Jon
I have 50 rupies I spend as below Spend remain 20 30 15 15 09 06 06 00 ----- ------- 50 51 why one more
Pls Help..... if f(x) =3x+2 what is the value of x whose image is 5
f(x) = 5 = 3x +2 x = 1
x=1
Bra
can anyone teach me how to use synthetic in problem solving
Mark
x=1
Mac
can someone solve Y=2x² + 3 using first principle of differentiation
ans 2
Emmanuel
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
which symbols
John
How to solve lim x squared two=4
why constant is zero
Rate of change of a constant is zero because no change occurred
Highsaint
What is the derivative of sin(x + y)=x + y ?
_1
Abhay
How? can you please show the solution?
Frendick
neg 1?
Frendick
find x^2 + cot (xy) =0 Dy/dx
Continuos and discontinous fuctions
integrate dx/(1+x) root 1-x square
Put 1-x and u^2
Ashwini
d/dxsinh(2xsquare-6x+4)
how
odofin
how to do trinomial factoring?
Give a trinomial to factor and I'll show you how
Bruce
i dont know how to do that either. but i really wants to know know how
Bern
Do either of you have a trinomial to present that needs factoring?
Bruce
5x^2+11x+2 and 2x^2+7x-4
Samantha
Thanks, so I'll show you how to do the 1st one and then you can try to do the second one. The method I show you is a general method you can use to factor ANY factorable polynomial.
Bruce
ok
Samantha
I have to link you to a document on how to do it since this chat does not have LaTeX
Bruce
Give me a few
Bruce
Sorry to keep you waiting. Here it is: ***mathcha.io/editor/ZvZkJUdrHpVHXwhe7
Bruce
You are welcome to ask questions if you have them. Good luck
Bruce
Can I ask questions with photos ?
Lemisa
If you want
Bruce
I don't see an option for photo
Lemisa
Do you know Math?
Lemisa
You have to post it as a link
Bruce
Can you tell me how to. Post?
Lemisa
Yes, you can upload your photo on imgur and post the link here
Bruce
thanks it was helpful
Samantha
I'm glad that was able to help you. Seriously, if you have any questions, please ask. That is not easy to understand at first so I anticipate you will have questions. It would be best to convince that you understood to attempt factoring the second trinomial you posted.
Bruce
it is use in differentiation for finding the limit if x turn to zero or infinity
let see
odofin
ok
odofin
What's this exactly?
Lemisa
calculus
Sal
approximation
Alejandro
can any body teach me calc3
Sal
what is limit?