<< Chapter < Page Chapter >> Page >
  • Determine the mass of a one-dimensional object from its linear density function.
  • Determine the mass of a two-dimensional circular object from its radial density function.
  • Calculate the work done by a variable force acting along a line.
  • Calculate the work done in pumping a liquid from one height to another.
  • Find the hydrostatic force against a submerged vertical plate.

In this section, we examine some physical applications of integration. Let’s begin with a look at calculating mass from a density function. We then turn our attention to work, and close the section with a study of hydrostatic force.

Mass and density

We can use integration to develop a formula for calculating mass based on a density function. First we consider a thin rod or wire. Orient the rod so it aligns with the x -axis, with the left end of the rod at x = a and the right end of the rod at x = b ( [link] ). Note that although we depict the rod with some thickness in the figures, for mathematical purposes we assume the rod is thin enough to be treated as a one-dimensional object.

This figure has the x and y axes. On the x-axis is a cylinder, beginning at x=a and ending at x=b.
We can calculate the mass of a thin rod oriented along the x -axis by integrating its density function.

If the rod has constant density ρ , given in terms of mass per unit length, then the mass of the rod is just the product of the density and the length of the rod: ( b a ) ρ . If the density of the rod is not constant, however, the problem becomes a little more challenging. When the density of the rod varies from point to point, we use a linear density function    , ρ ( x ) , to denote the density of the rod at any point, x . Let ρ ( x ) be an integrable linear density function. Now, 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 ] . [link] shows a representative segment of the rod.

This figure has the x and y axes. On the x-axis is a cylinder, beginning at x=a and ending at x=b. The cylinder has been divided into segments. One segment in the middle begins at xsub(i-1) and ends at xsubi.
A representative segment of the rod.

The mass m i of the segment of the rod from x i 1 to x i is approximated by

m i ρ ( x i * ) ( x i x i 1 ) = ρ ( x i * ) Δ x .

Adding the masses of all the segments gives us an approximation for the mass of the entire rod:

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

This is a Riemann sum. Taking the limit as n , we get an expression for the exact mass of the rod:

m = lim n i = 1 n ρ ( x i * ) Δ x = a b ρ ( x ) d x .

We state this result in the following theorem.

Mass–density formula of a one-dimensional object

Given a thin rod oriented along the x -axis over the interval [ a , b ] , let ρ ( x ) denote a linear density function giving the density of the rod at a point x in the interval. Then the mass of the rod is given by

m = a b ρ ( x ) d x .

We apply this theorem in the next example.

Calculating mass from linear density

Consider a thin rod oriented on the x -axis over the interval [ π / 2 , π ] . If the density of the rod is given by ρ ( x ) = sin x , what is the mass of the rod?

Applying [link] directly, we have

m = a b ρ ( x ) d x = π / 2 π sin x d x = cos x | π / 2 π = 1 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Consider a thin rod oriented on the x -axis over the interval [ 1 , 3 ] . If the density of the rod is given by ρ ( x ) = 2 x 2 + 3 , what is the mass of the rod?

70 / 3

Got questions? Get instant answers now!

We now extend this concept to find the mass of a two-dimensional disk of radius r . As with the rod we looked at in the one-dimensional case, here we assume the disk is thin enough that, for mathematical purposes, we can treat it as a two-dimensional object. We assume the density is given in terms of mass per unit area (called area density ), and further assume the density varies only along the disk’s radius (called radial density ). We orient the disk in the x y -plane, with the center at the origin. Then, the density of the disk can be treated as a function of x , denoted ρ ( x ) . We assume ρ ( x ) is integrable. Because density is a function of x , we partition the interval from [ 0 , r ] along the x -axis . For i = 0 , 1 , 2 ,… , n , let P = { x i } be a regular partition of the interval [ 0 , r ] , and for i = 1 , 2 ,… , n , choose an arbitrary point x i * [ x i 1 , x i ] . Now, use the partition to break up the disk into thin (two-dimensional) washers. A disk and a representative washer are depicted in the following figure.

Questions & Answers

How to do basic integrals
dondi Reply
write something lmit
ram Reply
find the integral of tan tanxdx
Lateef Reply
-ln|cosx| + C
Jug
discuss continuity of x-[x] at [ _1 1]
Atshdr Reply
Given that u = tan–¹(y/x), show that d²u/dx² + d²u/dy²=0
Collince Reply
find the limiting value of 5n-3÷2n-7
Joy Reply
Use the first principal to solve the following questions 5x-1
Cecilia Reply
175000/9*100-100+164294/9*100-100*4
Ibrahim Reply
mode of (x+4) is equal to 10..graph it how?
Sunny Reply
66
ram
6
ram
6
Cajab
what is domain in calculus
nelson
integrals of 1/6-6x-5x²
Namwandi Reply
derivative of (-x^3+1)%x^2
Misha Reply
(-x^5+x^2)/100
Sarada
(-5x^4+2x)/100
Sarada
oh sorry it's (-x^3+1)÷x^2
Misha
-5x^4+2x
Sarada
sorry I didn't understan A with that symbol
Sarada
find the derivative of the following y=4^e5x y=Cos^2 y=x^inx , x>0 y= 1+x^2/1-x^2 y=Sin ^2 3x + Cos^2 3x please guys I need answer and solutions
Ga Reply
differentiate y=(3x-2)^2(2x^2+5) and simplify the result
Ga
72x³-72x²+106x-60
okhiria
y= (2x^2+5)(3x+9)^2
lemmor
solve for dy/dx of y= 8x^3+5x^2-x+5
Ga Reply
192x^2+50x-1
Daniel
are you sure? my answer is 24x^2+10x-1 but I'm not sure about my answer .. what do you think?
Ga
24x²+10x-1
Eyad
eyad Amin that's the correct answer?
Ga
yes
Eyad
ok ok hehe thanks nice dp ekko hahaha
Ga
hahaha 😂❤️❤️❤️ welcome bro ❤️
Eyad
eyad please answer my other question for my assignment
Ga
y= (2x^2+5)(3x+9)^2
lemmor
can i join?
Fernando
yes of course
Jug
can anyone teach me integral calculus?
Jug
it's just the opposite of differential calculus
yhin
of coursr
okhiria
but i think, it's more complicated than calculus 1
Jug
Hello can someone help me with calculus one...
Jainaba
find the derivative of y= (2x+3)raise to 2 sorry I didn't know how to put the raise correctly
Ga Reply
8x+12
Dhruv
8x+3
okhiria
d the derivative of y= e raised to power x
okhiria
rates of change and tangents to curves
Kyaw Reply
how can find differential Calculus
Kyaw
Practice Key Terms 4

Get the best Calculus volume 1 course in your pocket!





Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask