What are Integrals and what do they tell you?

[thharrimw]

what are Integrals and what do they tell you?

 

[Pere Callahan]

It might help to have a look at an introductory textbook to get an idea. That's what they've been written for after all.

 

[HallsofIvy]

At about the time of the renaissance, there were two major problems being looked at by mathematicians: finding the tangent line to a curve (and so approximate some complicated function by a linear approximation) and finding the area of general two dimensional figure. The derivative solves the first of those problems and the integral solves the second. One reason why Newton and Leibniz are considered the founders of calculus, even though others had come up with formulas very similar to theirs is that they showed "the fundamental theorem of calculus", that these two problem are, in a sense, inverse to one another.

 

[lurflurf]

I like to think of them as linear operators.
L(a*f+b*g)=a*L(f)+b*L(g)
thus extenting a discrete opperator to a continuous one
L(lim f)=lim L(f)
thus say we can integtrate step functions
L(f)=sum f(xi*)[xi+1-xi]
we can integrate limits of step functions
using properties like
f<g ->L(f)<L(g)
L(a*f+b*g)=a*L(f)+b*L(g)
L(1)=b-a (ie the width of the inteval under consideration)

 

[thharrimw]

so integrals tell you the area of any given function

 

[Gib Z]

so integrals tell you the area of any given function

Well, sort of, yes.

 

[thharrimw]

how does knowing the area of a equation help you

 

[sutupidmath]

how does knowing the area of a equation help you

this doesn't really make sense!

-the definite integral
$$\int_a^{b}f(x)dx$$ is the area that is enclosed by the graph of the function f, and the x axis, in this case!

 

[Feldoh]

how does knowing the area of a equation help you

It's just just a representation of area in the physical sense. When we talk about integration in the simplest sense, we're talking about taking a lot of infinitely small portions of something and summing them together.

In calculus you think of an integral as the change in the value of a function whose derivative is given. Or you might think about it as the area to a particular equation. f(x)=sqrt(1-x^2) from -1 to 1, gives you an area of pi/2 for instance.

In physics you might think of integration as an infinite summation of a uniform line of charge on a particular point in space. Or as a method to find a change in position knowing only a velocity.

 

[makaveli]

how does knowing the area of a equation help you

Well, Say you're running and you want to see how far you have traveled, and you have a graph of your (Y)Velocity as (X)Time, but you have traveled in acceleration, so the normal approach of counting the squares under the line doesn't work. In this case you can:
-Proceed to count squares (good luck with that).

-whip out an integral and get it done like a man.

 

[Gib Z]

how does knowing the area of a equation help you

If you were asking, when is it beneficial to be able to evaluate the integral of a function?, then the answer would be - many cases =] Integrals let you show that the volume of a sphere is $$\frac{4}{3} \pi r^3$$ and its surface area is $$4\pi r^2$$, as well as many other things.

 

[mathwonk]

the first use of integrals was to compute the area under a parabola and the volume of a sphere.