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What exactly is integration and differentiation?

  1. Oct 16, 2011 #1
    in a non mathematical sense?

    My maths isn't very good which is why I need some simple explanation of what they are.

    What does integration do? What is its purpose and how does its purpose apply to the natural world? I know differentiation is the opposite process but what is it exactly?

    Where did it come from?
  2. jcsd
  3. Oct 16, 2011 #2


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    Integration of a real-valued function finds the area between the function and the x-axis, and differentiation finds the slope of a real-valued function at specific points.

    These operations are amazingly opposite operations, in the sense that
    the derivative wrt x of the integral of a function f from any constant to x is equal to f(x).

    Study elementary calculus to see how these things can be applied to physics. You're not going to get a good understanding of it if you don't know how to do it yourself.
  4. Oct 16, 2011 #3


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    differentiation is a way of telling "how fast something is changing". one example of this is the speedometer in a car. it is measuring "how fast your positon is changing in the direction you're travelling".

    now, normally, we think of speed as distance/time.

    but what about "instantaneous speed"? how do you tell how fast you're going "right now"?

    the distance/time formula does us no good, we just get 0/0 = ??????

    but....there IS a way, and it's kind of clever.

    suppose you're driving at a steady 60 miles an hour. then d/t should be 60, no matter how small "t" is. in other words:

    d/t = 60t/t....and we just "cancel the t's".

    that is the basic idea behind finding a limit, we see if (d(x+t)-d(x))/t is "well-behaved" (that is, we can approximate it better and better) even when t is very, very small.

    functions (like our distance = d(x)) that DO behave well under such circumstances, are called differentiable, which means that we can tell "where they're headed right NOW". not all functions are well-behaved, but many of the useful ones are.

    integration is sort of the "mirror process", given how fast things are changing "right now", can we tell what the effect of that change will be? that is, given how our speed varies over time, can we figure out our position? and, again for some, but not all functions, we can (if we know where we were, when we started). such functions for which we can "anti-differentiate" are called integrable.

    it turns out that we gain a bonus, that integration is also closely tied to what we call "length" (in one dimension), "area" (in two dimensions) and "volume" (in three dimensions). for higher dimensions (which are hard to imagine), the term "content" is often used.

    in the real world (i.e, in science and technology), we can use these features of "nice" functions to predict how things will behave. this is very useful. without calculus, we never would have made it to the moon and back (or made it very far in understanding electricty and magnetism).

    calculus arose from studying phenomenon that changed "smoothly" over time, gradually, or fluidly. like a billiard ball, rolling along the surface of a table (perhaps with a little spin), rather than like a flashing light that suddenly blinks off and on. in fact, it was the search for a way to capture this idea of smoothness, or continual change, that led us to unify rational and irrational numbers in one big happy family (fractions are too "grainy" to capture the fluid behavior we were after), the real numbers (and their complicated big brother, the aptly-named(?) complex numbers).

    none of this is terribly precise, but hopefully you get the idea.
  5. Oct 16, 2011 #4
    Last edited by a moderator: Apr 26, 2017
  6. Oct 16, 2011 #5
    You're driving your car. Your position is a function of time ... at time zero you're home; at time 1 hour you're 50 miles down the freeway; at time 2 hours you're 100 miles down the freeway.

    Your speedometer gives your instantaneous speed ... that's the first derivative of your position function.

    The total distance travelled is the integral of your position function.

    It's that simple.

    (edit) A little more.

    Say you drive at a varying speed. At any moment, your speedometer tells you how fast you're going at that instant. If you keep track of your speedometer, can you figure out how far you've gone? Well, you could do this: Every minute, look at your speedometer and use "distance = rate times time" to see how far you went in the past minute. That's not completely accurate because your speed varied during that minute ... but your minute-by-minute sample isn't too far from the true total distance.

    So you divide up your trip into 1-minute segments, add up all the "distance = rate times time" calculations, and you have an estimate of the total distance travelled.

    Now you could get a better estimate by sampling your speedometer ever 10 seconds and doing the same "sum of the distance = rate times time" calculations. And you could do even better by sampling every second.

    If you followed that, then you understand the fundamental theorem of calculus, which says you can determine the total distance travelled by doing "sum of distance = rate times time" over smaller and smaller sample intervals.
    Last edited: Oct 16, 2011
  7. Oct 18, 2011 #6
    Welcome student, :smile:

    geometry gives you an idea of integration "also in a math sense" . Imagine, draw a rectangle: base = x and height = y = k. Math, geometry tells you that: Area = B x h,

    A rectangle = x * y
    now imagine, draw a "rectangle" with an irregular top: base = x , height = y is not k but equals x² . Calculus [integration] tells you that

    Area "rectangle" = x³/ 3

    a great invention
    Last edited: Oct 18, 2011
  8. Oct 18, 2011 #7
    This is the highly non rigorous explanation but it is a good start to understand
    how simple the ideas really are underneath all of the equations, proofs and theorems.

    Calculus is the Mathematics of Change

    Integration is Multiplication, when one of the multiplicands is changing

    Differentiation is the Instantaneous Slope of the tangent line to the function at a given point.
    Instantaneous Slope = Average Slope between two points on the function
    as the 2 points get closer and closer together in the Limit as delta x=>0.
    delta x = the distance between the two x values of the points = x2 - x1.

    Now for your questions............

    What does integration do?
    It allows you to multiply in complex situations to solve difficult problems

    What is its purpose and how does its purpose apply to the natural world?
    Math is the language of the natural world.
    Calculus is just one part of that language.
    English, German, Spanish, etc. are hopelessly inaccurate and verbose to
    describe the complex phenomena of science. Thus we invented Math.

    I know differentiation is the opposite process but what is it exactly?
    That Differentiation and Integration are Inverse [opposite] Functions
    is the Fundamental Theorem of Calculus.

    Where did it come from?
    This is a theological question for which Science and Mathematics have
    no comment.
    Last edited: Oct 18, 2011
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