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This figure has two images. The first is labeled “a” and is a circle with radius r. The center of the circle is labeled 0. The circle also has the positive x-axis beginning at 0, extending through the circle. The second figure is labeled “b”. It has two concentric circles with center at 0 and the x-axis extending out from 0. The concentric circles form a washer. The width of the washer is from xsub(i-1) to xsubi and is labeled delta x.
(a) A thin disk in the xy -plane. (b) A representative washer.

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

A i = π ( x i ) 2 π ( x i 1 ) 2 = π [ x i 2 x i 1 2 ] = π ( x i + x i 1 ) ( x i x i 1 ) = π ( x i + x i 1 ) Δ x .

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 * ( x i + x i 1 ) / 2 to approximate the average radius of the washer. We obtain

A i = π ( x i + x i 1 ) Δ x 2 π x i * Δ x .

Using ρ ( x i * ) to approximate the density of the washer, we approximate the mass of the washer by

m i 2 π x i * ρ ( x i * ) Δ x .

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

m = i = 1 n m i i = 1 n 2 π x i * ρ ( x i * ) Δ x .

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

m = lim n i = 1 n 2 π x i * ρ ( x i * ) Δ x = 0 r 2 π x ρ ( x ) d x .

We summarize these findings in the following theorem.

Mass–density formula of a circular object

Let ρ ( x ) be an integrable function representing the radial density of a disk of radius r . Then the mass of the disk is given by

m = 0 r 2 π x ρ ( x ) d x .

Calculating mass from radial density

Let ρ ( x ) = x represent the radial density of a disk. Calculate the mass of a disk of radius 4.

Applying the formula, we find

m = 0 r 2 π x ρ ( x ) d x = 0 4 2 π x x d x = 2 π 0 4 x 3 / 2 d x = 2 π 2 5 x 5 / 2 | 0 4 = 4 π 5 [ 32 ] = 128 π 5 .
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Got questions? Get instant answers now!

Let ρ ( x ) = 3 x + 2 represent the radial density of a disk. Calculate the mass of a disk of radius 2.

24 π

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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 ( kg · m 2 / s 2 ) .

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 ( x ) 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 [ a , b ] and estimate the work done over each subinterval. So, for i = 0 , 1 , 2 ,… , n , let P = { x i } be a regular partition of the interval [ a , b ] , and for i = 1 , 2 ,… , n , choose an arbitrary point x i * [ x i 1 , x i ] . 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 ( x i * ) to approximate the force. The work done over the interval [ x i 1 , x i ] , then, is given by

Questions & Answers

find the domain and range of f(x)= 4x-7/x²-6x+8
Nick Reply
find the range of f(x)=(x+1)(x+4)
Jane Reply
-1, -4
Marcia
That's domain. The range is [-9/4,+infinity)
Jacob
If you're using calculus to find the range, you have to find the extrema through the first derivative test and then substitute the x-value for the extrema back into the original equation.
Jacob
Good morning,,, how are you
Harrieta Reply
d/dx{1/y - lny + X^3.Y^5}
mogomotsi Reply
How to identify domain and range
Umar Reply
hello
Akpevwe
He,,
Harrieta
hi
Dr
hello
velocity
I only talk to girls
Dr
women are smart then guys
Dr
Smarter
Adri
sorry
Dr
hi adri ana
Dr
:(
Shun
was up
Dr
hello
Adarsh
is it chatting app?.. I do not see any calculus here. lol
Adarsh
Find the arc length of the graph of f(x) = In (sinx) on the interval [Π/4, Π/2].
mukul Reply
Sand falling freely from a lorry form a conical shape whose height is always equal to one-third the radius of the base. a. How fast is the volume increasing when the radius of the base is (1m) and increasing at the rate of 1/4cm/sec Pls help me solve
ade
show that lim f(x) + lim g(x)=m+l
BARNABAS Reply
list the basic elementary differentials
Chio Reply
Differentiation and integration
Okikiola Reply
yes
Damien
proper definition of derivative
Syed Reply
the maximum rate of change of one variable with respect to another variable
Amdad
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)
Inembo Reply
what is calculus?
BISWAJIT Reply
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?
Geoffrey Reply
what is x and how x=9.1 take?
Pravin Reply
what is f(x)
Inembo Reply
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
Adri
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).
Adri
I dont understand what you wanna say by (A' n B^c)^c'
Adri
(A' n B (rise to the power of c)) all rise to the power of c
Inembo
Aaaahh
Adri
Ok so the set is formed by vectors and not numbers
Adri
A vector of length n
Adri
But you can make a set out of matrixes as well
Adri
I I don't even understand sets I wat to know d meaning of all d symbolsnon sets
Inembo
Wait what's your math level?
Adri
High-school?
Adri
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
Adri
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
Adri
I would say 24
Adri
Offer both
Adri
Sorry 20
Adri
Actually you have 40 - 4 =36 who offer maths or physics or both.
Adri
I know its 20 but how to prove it
Inembo
You have 32+24=56who offer courses
Adri
56-36=20 who give both courses... I would say that
Adri
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
Practice Key Terms 4

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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