What are Integrals and what do they tell you?

• thharrimw
In summary, integrals are mathematical tools used to find the area under a curve or the volume of a solid shape. They were developed around the time of the Renaissance to solve two major problems in mathematics: finding tangent lines to curves and finding the area of two-dimensional figures. They are considered the foundation of calculus and have many practical applications in fields such as physics and engineering.
thharrimw
what are Integrals and what do they tell you?

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.

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.

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)

so integrals tell you the area of any given function

thharrimw said:
so integrals tell you the area of any given function

Well, sort of, yes.

thharrimw said:

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!

thharrimw said:

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.

Last edited:
thharrimw said:

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.

thharrimw said:

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.

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

What are Integrals?

Integrals are mathematical tools used to calculate the total area under a curve or between two curves. They are also used to find the volume of a three-dimensional shape and to solve various real-world problems in physics, engineering, and economics.

How do you calculate Integrals?

To calculate an integral, you need to find the antiderivative of the function being integrated and then evaluate it at the upper and lower limits of the integral. This is typically done using integration techniques such as substitution, integration by parts, or partial fractions.

What do Integrals tell you?

The value of an integral represents the sum of all the small changes in the function over a given interval. In other words, it tells you the net effect of the function over that interval. For example, the integral of a velocity function would give you the total distance traveled over a given time period.

What are the different types of Integrals?

The two main types of integrals are definite and indefinite integrals. Definite integrals have specific upper and lower limits, while indefinite integrals do not. Other types of integrals include improper integrals, line integrals, and surface integrals.

What applications do Integrals have?

Integrals have a wide range of applications, including finding the area and volume of shapes, determining the total displacement or distance traveled by a moving object, and solving optimization problems in economics and physics. They are also used in calculus, physics, and engineering to model and solve various real-world problems.

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