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 Sirajuddin, David Sirajuddin, David Itcanbeshown.com Itcanbeshown.com
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How is integration useful in physics? Integration is addition: we just change the notation [Actually, integration is given a different notation because.
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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
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.
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)
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
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