What is an integral and how do you solve it?

[Krab19]

Hey guys! I have absolutely no experience with integrals, but i find them interesting. I do know that they represent the area between the function and the x-axis, but not much else. I was wondering if anyone could give me a simple explanation of what an integral is and how to actually solve one.

Thank you in advance

 

[Office_Shredder]

You're asking for half of a calculus class basically

 

[iceblits]

if you're asking for simple computation of integrals...then all you have to do is look up the form of the integral on a table and then look at the solution. It's generally a good idea to think of integration not just as the area under a curve but also as a sum of continuous "things". Integration takes practice, but you can always use a calculator (ti-83 or better) to get an integral...

 

[Nebuchadnezza]

http://tutorial.math.lamar.edu/Classes/CalcI/IntegralsIntro.aspx

 

[Krab19]

http://tutorial.math.lamar.edu/Classes/CalcI/IntegralsIntro.aspx

Thank you very much :)

 

[dalcde]

how to actually solve one.

You don't actually "solve" an integral. You EVALUATE an integral. (I know I'm not helping, but I'm irritated by people who keep saying "solve" an integral or "solve" a derivative while there is nothing to actually "solve")

 

[bp_psy]

If you are only interested on seeing computation try:
http://www.youtube.com/user/khanacademy#g/c/19E79A0638C8D449

 

[HallsofIvy]

You don't actually "solve" an integral. You EVALUATE an integral. (I know I'm not helping, but I'm irritated by people who keep saying "solve" an integral or "solve" a derivative while there is nothing to actually "solve")

And I'll bet you are really annoyed at our British friends who use the phrase "doing a sum" to mean any kind of mathematical calculation!

 

[Mute]

And I'll bet you are really annoyed at our British friends who use the phrase "doing a sum" to mean any kind of mathematical calculation!

Ugh. That's even more annoying than "maths".

 

[mathwonk]

an integral is a certain limit. you can read up what this limit is in any calculus book and you should. doing, or solving or whatever, to the integral involves finding or evaluating that limit.

there are often no practical ways to find this limit, as is usual with real life limits. But in many interesting cases these limits can be found in various ways. Archimedes managed to find some, enough to compute the volume of a ball.

The calculus book by Apostol is excellent for giving several interesting and non trivial examples of integrals which can be computed by hand.

A few hundred years ago it was noticed that if the integral is looked at as defining a function, then there is a certain differential equation satisfied by this function. this is called the fundamental theorem of calculus

That implies that if we can solve the differential equation, we can also evaluate the integral. In many many specific cases these solutions, called "antiderivatives" have been written down in tables and can be looked up.

But it is important to understand that in general there is no known explicit solution to the differential equation, and in fact the only way to define a solution, is to use the limit definition of the integral. e.g. even the integral of a simple function like cos(x^2) apparently cannot be evaluated by the FTC.

So books that discuss differentiation first and then use only the fundamental theorem to evaluate integrals do a disservice, since those students never appreciate what an integral is, nor the fact that most of the time in the real world, the theorem is of no use.

Again, if you want to understand integrals, and calculus in general, read a book like apostol.

 

[Mark44]

You don't actually "solve" an integral. You EVALUATE an integral. (I know I'm not helping, but I'm irritated by people who keep saying "solve" an integral or "solve" a derivative while there is nothing to actually "solve")

I'm with you all the way on this. It's sort of akin to asking someone to "solve" 3 + 2, or "solve the equation" 5x + 3.

 

[Mark44]

I do know that they represent the area between the function and the x-axis, but not much else.

This sentence is unclear. I know what you mean, but I had to think about it a bit.

The uncertaintly lies in what the phrase "but not much else" applies to. What you wrote could be interpreted in two ways:
1)"(integrals) represent the area between the function and the x-axis, but not much else." or
2) "I know integrals represent the area between a function's graph and the x-axis, but that's all I know."

You might think I'm being picky here, but mathematics is very precise, and to do well at it, it helps to think with precision.

 

[mathwonk]

there may be room for interpretation here in the matter of language. e.g. to evaluate the definite integral of f from a to b, one usually proceeds by solving the differential equation g' = f on that interval. Some people even call g an "integral" of f. Unless there is reason to think the language is the problem, i usually try to focus on the ideas rather than obsess on "correct" terminology.

Two good places to learn what the idea of integral means are the first 150 pages of apostol, as mentioned above, and the first part of michael comenetz's book the elements of calculus.

 

[paulfr]

What is an Integration ?
It is basically "Multiplication"
That is why Area is one interpretation.
But Volume or Work calculations are also done by integration.
You integrate anytime you need to multiply when one of the multiplicands is changing.
Remember that Calculus is the Mathematics of Change.

Repetitive Addition is another interpretation because that is what Multiplication is.
Notice the integral symbol looks like an "S" as it is meant to be Summation.

Think of it this way.
If f(x) is y=3 and you are integrating from x=2 to x=4
You get the area of 3 x 2 = 6
Now suppose f(x) is more complex, like f(x) = sin x.
Now you need to integrate by chopping up the area into long thin vertical strips and
finding the areas of each of them and summing them all up.
The limit comes about only to increase accuracy.
In fact Numerical Methods approximate the calculation without the Limit of dx => 0

Cheers

 

[LaurieAG]

A differential in basic integral calculus is when you apply n*x^(n-1) to an equation in the form of A*x^2 + B*x^(1) + C*x^(0) = 0.

The integral is just the reverse of this process so the integral of 2*A*x + B + 0 = 0 is equivalent to the original form above.

As the differential is also the area between the original curve and the x-axis you will find that integrals often have limits that just define where this area is measured.

 

[Krab19]

Sorry I have been fairly busy the past week, but thank you all for your help, I now have a good general idea of integrals :)