What is integration?

Question

I keep finding mathematical formulas which include integration in them as a fundamental part. However i don't know what integration does, how it works or anything, could someone please explain the basics to me. Cheers

Answer 1

If you know differentiation then it is very short:
Integration is the operation inverse to derivation.

It is a little more complicated than differentiation and many times does not give a single answer.

Example:

If y = x then the derivative y' = 1

The integral of 1 is x, because it is the inverse operation. BUT, x + 1 is also the integral of 1 because the derivative of x + 1 is also 1.

So in general the integral of 1 will be written as x + C, where C is a numerical constant, usually found by applying an initial condition, lie y(0) = 3.

If you have y(x) = x + C with y(0)=3 you can calculate C.

Answer 2

Integration is used to calculate "areas under graphs", and can also be thought of as the reverse to differentiation.

Thus if f(x)=x^2, then the derivative of f(x), denoted by f'(x)=2x. Hence, the integral of 2x is (x^2) +c, where c is the constant of integration (this is needed because the derivative of a constant is zero, and hence (x^2) + 5 say, would have the same derivative as x^2.)

Integration can be much more complicated than differentiation, however! If you try to integrate a function like log(x), you will find it difficult to think of a function whose derivative is log(x) (although you can integrate log(x) using a method known as integration by parts)

Try having a look in an AS math book (used to be in P2 on the edexcel syllabus, which may be C2 or C3 now they've changed the syllabus). Integration/differentiation isn't taught below the 6th form in schools, so you should find an introduction to both integration and differentiation in AS math text books.

Answer 3

Integration is the opposite of differentiation. Together they make up the subject of calculus.

Differentiation (when you see something like dy/dx) describes how a quantity (y) varies as a function of another quantity x.

Integration is the reverse of this process. For example, the distance travelled (s) is a function of time (t). So, when you calculate ds/dt you get an expression for how s is changing as a function of time. In everyday language you have worked out the velocity or ds/dt = v.

Now, lets say you know v and want to know s at a particular time. Then you integrate ds/dt with respect to time. The instruction to integrate is written as a funny looking elongated S.

Integration is also used to calculate the area under a curve, the volume swept out by a rotating shape and a host of others.

Answer 4

Integration is the opposite of differentiation.
When differentiating - 3x^2 becomes 6x^3 (so the index multiplys the number in front of the x, and the power goes up by 1)
When integrating - 6x^3 becomes 3x^2 (the index is reduces by 1 and the number in front of the x is divided by the new power.

Differentiation is used to find the gradient of a line. (how steep it is)
Integration is used to find the area under a line, or between lines.

These can also be used to find distance, speed and acceleration if you have an equation which represents one of the three quantities above.

Answer 5

Integration basically works out the area between a curve and the x-axis.

But you're probably trying to read stuff that is too high a level for you. The thing about maths is that stuff you'll understand in just a few months, often looks alien to you beforehand.

Answer 6

noun 1. an act or instance of combining into an integral whole.
2. an act or instance of integrating a racial, religious, or ethnic group.
3. an act or instance of integrating an organization, place of business, school, etc.
4. Mathematics. the operation of finding the integral of a function or equation, esp. solving a differential equation.
5. behavior, as of an individual, that is in harmony with the environment.
6. Psychology. the organization of the constituent elements of the personality into a coordinated, harmonious whole.
7. Genetics. coadaptation (def. 2).


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[Origin: 1610–20; integrate + -ion; cf. L integrātiō renewal]


—Synonyms 1. combination, blending, fusing.

Answer 7

This is a hard question to sum up in this little text box.
Integration is a method that originated for calculating areas, it is perhaps better if you look it up in wikipedia.
Or get an intro to calculus book. It really is a topic that requires more than a two second explanation

: )

Answer 8

integration adds all the area under a curve up. if you have something simple like y=3, and you integrate from x1=0 to x2=3, the area will be nine.
y=3 acts like a height, and x2-x1 is like a width.

so, you just add all the area under a curve.

Answer 9

Integration by way of factors. The functionality is a manufactured from 2 purposes: (x) * (cosx) call one in all them "u" and the different "dv" pick the "u" so as that its spinoff simplifies u = x du = a million*dx dv = cosx v = sinx this is for the formula setup, it somewhat is u*v - indispensable (vdu) =xsinx - indispensable(sinx * a million * dx) =xsinx - (-cosx) + C = xsinx + cosx + C

Answer 10

The answer to this question is more complex...

For start check this out:
http://en.wikipedia.org/wiki/Integral