Motivation behind the integral?

In summary: I should care about the integral itself, instead of just the various measures that it leads to. What is the difference?
  • #1
holezch
251
0
Hi, this has been bugging me for a long time: I am lacking a philosophical concept or just a general motivation behind the concept of an integral. To me, "computing the area under a graph" seems too specialized to me, especially how our pictorial representation of a function could have been arbitrary. When I ask why I should care about the integral at all, I'm usually given answers like "to find the volume of a swimming pool.." , " to build MRI scans.." and I am unsatisfied with these answers. When I study algebra (vector spaces, rings ) it has a nice "beautiful picture" behind it.. I do not find this same intrinsic interest when I look at integration, can someone help motivate or enlighten me? thank you
 
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  • #2
Look up "integral applications" with Google or Bing or your favorite search engine. You will be overwhelmed with examples.
 
  • #3
Integration is, in essence, the inverse of differentiation. The Fundamental Theorem of Calculus says, in part,
[tex]f(x) = \frac{d}{dx}\int_a^x f(t)dt[/tex]
 
  • #4
holezch said:
Hi, this has been bugging me for a long time: I am lacking a philosophical concept or just a general motivation behind the concept of an integral. To me, "computing the area under a graph" seems too specialized to me, especially how our pictorial representation of a function could have been arbitrary. When I ask why I should care about the integral at all, I'm usually given answers like "to find the volume of a swimming pool.." , " to build MRI scans.." and I am unsatisfied with these answers. When I study algebra (vector spaces, rings ) it has a nice "beautiful picture" behind it.. I do not find this same intrinsic interest when I look at integration, can someone help motivate or enlighten me? thank you

In one sense you are finding out different measures (length, area, volume etc) that have non-linear parts. It generalizes all the notions of measure to the point where you can calculate appropriate measures of any kind of non-linear line (ie a curve of some sort).

Add into the equation vector calculus and you can compute answers involving vector concepts such as the sum of "projections" of a vector field on to some plane. Again just like the above example with measures: we are generalizing to calculate examples that aren't simply linear (in vector calculus our "linear" object is a plane).

The other thing is that in mathematical modeling, we don't know the explicit solution to some system, but instead we know how it changes instantaneously. Calculus provides the method to go from our initial definition of "infinitesimal" change to either an explicit model of our system, or if the analytical solution is too complicated or can't be found, a numerical simulation is used to model the system as accurately as possible.

In high school we were mostly taught about different measures like length, area, volume and so on, but all of these measures involved straight lines (ie linearity). So generalizing this to non-linear situations means the invention of calculus.
 
  • #5
mathman said:
Look up "integral applications" with Google or Bing or your favorite search engine. You will be overwhelmed with examples.

I don't want to see more "integral applications" , that is the reason why I made this thread
Mark44 said:
Integration is, in essence, the inverse of differentiation. The Fundamental Theorem of Calculus says, in part,
[tex]f(x) = \frac{d}{dx}\int_a^x f(t)dt[/tex]

I was going with this for a while, but not all functions that are integrable have primitives, so I don't think this is general enough
chiro said:
In one sense you are finding out different measures (length, area, volume etc) that have non-linear parts. It generalizes all the notions of measure to the point where you can calculate appropriate measures of any kind of non-linear line (ie a curve of some sort).

Add into the equation vector calculus and you can compute answers involving vector concepts such as the sum of "projections" of a vector field on to some plane. Again just like the above example with measures: we are generalizing to calculate examples that aren't simply linear (in vector calculus our "linear" object is a plane).

The other thing is that in mathematical modeling, we don't know the explicit solution to some system, but instead we know how it changes instantaneously. Calculus provides the method to go from our initial definition of "infinitesimal" change to either an explicit model of our system, or if the analytical solution is too complicated or can't be found, a numerical simulation is used to model the system as accurately as possible.

In high school we were mostly taught about different measures like length, area, volume and so on, but all of these measures involved straight lines (ie linearity). So generalizing this to non-linear situations means the invention of calculus.
Thank you, this kind of motivation is what I would like to hear.. I also realized that it's interesting intrinsically because it's just another function you can somewhat easily form out of old functions, and satisfies interesting properties such as it always being continuous.. et c

is integration a way to "measure" a set?thanks everyone
 
  • #6
holezch said:
Thank you, this kind of motivation is what I would like to hear.. I also realized that it's interesting intrinsically because it's just another function you can somewhat easily form out of old functions, and satisfies interesting properties such as it always being continuous.. etc
is integration a way to "measure" a set?
thanks everyone

I haven't done a graduate course in analysis, but there are actually many different types of integrals.

The one people are taught early on in calculus classes is known as the Riemann integral. Its basically represented as a lot of rectangles that have a height of some function value, and basically the width of the boxes approaches zero without actually being zero (since 0 x a = 0 for any a).

The next version of integration is known as the Lebesgue integral. This involves more math including sets and measures.

In a way you're analogy of "measuring a set" can have some validity, but without being more specific I can't say whether the idea in your head is the same as the idea in mine.
 
  • #7
Generally, speaking, things that in, nice, easy, constant situations, involve division, such as finding the slope of a line, the speed of an object, or the weight or mass density of something, become the derivative when things are not so nice.

Conversely, problems that involve multiplication, such as finding area or volume, finding distance moved given the speed, finding the weight given the density, become integral problems when we do not have constant height, speed, density, etc.
 
  • #8
holezch said:
I was going with this for a while, but not all functions that are integrable have primitives, so I don't think this is general enough
All integrable functions have entire families of primitives, namely if f(x) is integrable, then one primitive of f(x) is
[tex]F(x) = \int_k^x f(t) dt[/tex]
where k is an arbitrary constant. I believe you may be referring to the fact that most primitives cannot be written as finite combinations and compositions of the few elementary functions that we find useful in the most common idealistic situations, which should not be surprising, as the vector space of functions over R is infinite dimensional. The natural logarithm is defined as exactly this type of function, a special primitive of 1/x.
We find quite a few more functions useful in addition to the elementary functions, and many of them are defined using integrals. See Special functions. The only reason they are not taught at basic high school level is that they are only used in very specific circumstances or in advanced theory.

is integration a way to "measure" a set?
Yes. You should check out a basic text on measure theory; it is used as a basis for probability and statistics.
 
  • #9
slider142 said:
All integrable functions have entire families of primitives, namely if f(x) is integrable, then one primitive of f(x) is
[tex]F(x) = \int_k^x f(t) dt[/tex]
where k is an arbitrary constant. I believe you may be referring to the fact that most primitives cannot be written as finite combinations and compositions of the few elementary functions that we find useful in the most common idealistic situations, which should not be surprising, as the vector space of functions over R is infinite dimensional. The natural logarithm is defined as exactly this type of function, a special primitive of 1/x.
We find quite a few more functions useful in addition to the elementary functions, and many of them are defined using integrals. See Special functions. The only reason they are not taught at basic high school level is that they are only used in very specific circumstances or in advanced theory.


Yes. You should check out a basic text on measure theory; it is used as a basis for probability and statistics.

thank you, I have to go to work now but I will look into this


Not all integrable functions have primitives ( i.e. step functions, derivatives cannot have pointwise discontinuities ), a primitive is guaranteed if the function f is continuous
 
  • #10
holezch said:
thank you, I have to go to work now but I will look into this


Not all integrable functions have primitives ( i.e. step functions, derivatives cannot have pointwise discontinuities ), a primitive is guaranteed if the function f is continuous

True, I was thinking of primitive as just a function that agreed with the definite integral, but I see that it needs to be differentiated back to the integrand.
 

What is the purpose of using an integral?

The purpose of using an integral is to find the area under a curve or the value of a function over a certain interval. It is also used to solve problems involving change, such as finding the distance traveled or the amount of work done.

What is the difference between definite and indefinite integrals?

A definite integral has specific limits of integration and gives a numerical value as the result. An indefinite integral does not have limits of integration and gives a general function as the result.

How is the motivation behind the integral related to the concept of accumulation?

The motivation behind the integral is closely related to the concept of accumulation. The integral represents the accumulation of infinitely small quantities to find the total quantity. It can also be thought of as the sum of all the infinitely small changes in a quantity over a given interval.

What are some real-world applications of the integral?

The integral has many real-world applications, such as calculating areas and volumes in engineering and architecture, determining the rate of change in physics and economics, and predicting the growth of populations in biology and sociology.

Can the integral be used to solve optimization problems?

Yes, the integral can be used to solve optimization problems. By finding the total area or volume under a curve, it can help determine the optimal value of a variable in a given situation, such as maximizing profit or minimizing cost.

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