how is integration useful in physics? integration is addition: we just change the notation...

How is integration useful in physics?

Integration is addition: we just change the notation

[Actually, integration is given a different notation because it is a special kind of sum (we will get to that in a bit)]

Motivation: Often, we will have an equation defined for a quantity of interest (e.g. magnetic field, moment of inertia, volume) only for a point, or for some shape. If we have more complex shapes, we can find the overall quantity of the system by adding (or integrating) up contributions from our point/shape formulation

Addition Integration

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Example: Calculating Area

Consider a function for 0 ≤ x ≤ 2

What is the area scribed by the curve?

)sin()( 2xxf



We can sum the areas of smaller shapes (e.g. rectangles) to find it

Area

f(xi) = height, x = width We have taken sample values at 4 points here, but this is not a good

approximation How do we improve our approximation? Take more sample values, and

decrease the width

...)()()( 2211

4

1

xxfxxfxxfAi

ii



Try 5 values of f(x):

Area

f(xi) = height, x = width This is a better estimate! What happens if we take more samples

and continue to decease the width x?

...)()()( 2211

5

1

xxfxxfxxfAi

ii



We find as we sample more points, and continue to decrease the width, we find an exact area

Area

f(xi) = height, x = width, N = number of points sampled

...)()()( 22111

xxfxxfxxfAN

iii



Specifically, we note we get an exact area in a special limit:

x 0 Number of points sampled in f(xi), N ∞, i.e. we sample every value of x

In this limit, notation is changed

So that, in all, can write

We sometimes say that we “add up” differential elements Here, a differential element is one of our samples: f(xi)x. A differential element of length is

defined as x = dx (this word is what the ‘d’ is for, it does not mean a derivative at all), specifically an element dx is defined to have zero length, x 0.

The end! That is all integration is

N

i

x dx)( ixf )(xf

N

ii dxxfxxf

1

)()(


Generalized Area Calculation

Area of any shape:

It is possible to rewrite this if a functon f is given

Notice how integrating over dy is not useful here (see right) without some manipulation [f(y) is not one-to-one]

In cylindrical coordinates:

dxdydAAAi

i


dxxfA )( dyyfA )(or

rdrddrdr

yxdxdyA

),(

),(

(chain rule)

Example: Calculating Volume

This generalizes to volumes directly, consider a paraboloid:

This time, instead of adding up areas, we add up volumes Vi = Aixi


Example: Calculating Volume

This generalizes to volumes directly, consider a paraboloid:

This time, instead of adding up areas, we add up volumes Vi = Aixi

As x 0 (or V 0), we set x = dx (or V = dV), and we calculate an exact volume in our sum (integral) when we sample N ∞ points

(for this case, adding up disc volumes is convenient, width dx, area A)


N

ii dVVV

N

ii dxxAxAV )(or

(triple integral) (single integral)

Generalized Volume Calculation

Volume of any shape:

It is possible to rewrite this if the area A can be found as a function of x, y, or z…for the example on the right

we can sum discs in the x direction, or cylindrical shells in the y and z directions

In other coordinate systems

dxdydzdVVVi

i


dxxydxxAV 2)]([)(

dyyyxdrrhdyyAV )(2"2")(

(Cylindrical)

or dxxz 2)]([Discs:

Cyl. Shells:or

dzzzx )(2

(Spherical) dzrdrddxdydz ddrdrdxdydz sin2

Example: Stalactites/Stalagmites

Stalactites/stalagmites are found in limestone caves. They are formed from rainwater percolating through the soil to cave ceilings, where they dissolve limestone and “pull” it downward (Shown here, ~ 500,000 years of formation)

What is the mass of a single stalactite? Let us model it as the following:

Stalactite is ~ cone of height H, radius R Its mass density [mass/volume] changes linearly with depth z

(z) = Az + B, where A,B are constants (can be negative)

Then, the mass Is this reasonable? Check units: dV = [volume], (z) =

[mass/volume], then

Notice that since the density is different at each z, we cannot just multiply some density by the total volume, but instead have to add up the product of each volume element at every z with the density at every z (i.e. integrate)

Volume

zdVm )(

Carlsbad Caverns, NM [Wayfaring Travel Guide]

massvolume

massvolumezdVm )(


N

i

N

iiii zdVVzmm )()(

dVdmm

Example: Stalactites/Stalagmites

Geometry is convenient to add up discs Disc radius = r (this changes with z!)

We can take r to be x, or y (see figure), either is equivalent Disc height/thickness = dz each disc has a volume dV = dAdz = y2dz Where dA = area of the circular face of a disc of radius r = y We put a ‘d’ in front of dz, dA, and dV just to mean ‘differential’ This is just the language, the letter does not do anything (not a

derivative) Radius r is the same as y, we notice y changes linearly with z,

If it helps, turn the picture on its side to find this equation

So that,

RzH

Rzy )(

z

x

y

R

RzH

Rzy )(


22

RzH

RydV

H

H

dzBAzRzH

Rm

zdzyzdAdzzdVm

0

2

0

2 )()()()()(

Think about this! This gives us the distance in y measured from the center of the cone (a radius). Not only for one value, this function gives us the radius at any value of z we want

how is integration useful in physics? integration is addition: we just change the notation...

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width x

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