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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 AdditionIntegration Sirajuddin, David Itcanbeshown.com

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Example: Calculating Area Consider a function for 0 x 2 What is the area scribed by the curve? Sirajuddin, David Itcanbeshown.com

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Example: Calculating Area We can sum the areas of smaller shapes (e.g. rectangles) to find it Area f(x i ) = 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 Sirajuddin, David Itcanbeshown.com

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Example: Calculating Area Try 5 values of f(x): Area f(x i ) = height, x = width This is a better estimate! What happens if we take more samples and continue to decease the width x? Sirajuddin, David Itcanbeshown.com

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Example: Calculating Area We find as we sample more points, and continue to decrease the width, we find an exact area Area f(x i ) = height, x = width, N = number of points sampled Sirajuddin, David Itcanbeshown.com

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Example: Calculating Area Specifically, we note we get an exact area in a special limit: x 0 Number of points sampled in f(x i ), 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(x i ) 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 Sirajuddin, David Itcanbeshown.com

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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: Sirajuddin, David Itcanbeshown.com or (chain rule)

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Example: Calculating Volume This generalizes to volumes directly, consider a paraboloid: This time, instead of adding up areas, we add up volumes V i = A i x i Sirajuddin, David Itcanbeshown.com

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Example: Calculating Volume This generalizes to volumes directly, consider a paraboloid: This time, instead of adding up areas, we add up volumes V i = A i x i 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) Sirajuddin, David Itcanbeshown.com or (triple integral)(single integral)

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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 Sirajuddin, David Itcanbeshown.com (Cylindrical) or Discs: Cyl. Shells: or (Spherical)

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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) Carlsbad Caverns, NM [Wayfaring Travel Guide] Sirajuddin, David Itcanbeshown.com

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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 = y 2 dz 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, Sirajuddin, David Itcanbeshown.com 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

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