Introduction to Integration

Integration is a way of adding slices to find the whole.

Integration can be used to find areas, volumes, central points and many useful things. But it is easiest to start with finding the area under the curve of a function like this:

integral area = ?
What is the area under y = f(x) ?


We could calculate the function at a few points and add up slices of width Δx like this (but the answer won't be very accurate):

 integral area big deltax

We can make Δx a lot smaller and add up many small slices (answer is getting better):


 integral area small deltax

And as the slices approach zero in width, the answer approaches the true answer.

We now write dx to mean the Δx slices are approaching zero in width.

 integral area dx

That is a lot of adding up!

But we don't have to add them up, as there is a "shortcut". Because ...

... finding an Integral is the reverse of finding a Derivative.

(So you should really know about Derivatives before reading more!)

Like here:

Example: What is an integral of 2x?

integral vs derivative


We know that the derivative of x2 is 2x ...


... so an integral of 2x is x2

You will see more examples later.


The symbol for "Integral" is a stylish "S"
(for "Sum", the idea of summing slices):

 integral notation

After the Integral Symbol we put the function we want to find the integral of (called the Integrand),

and then finish with dx to mean the slices go in the x direction (and approach zero in width).

And here is how we write the answer:

integral of 2x dx = x^2 + C

Plus C

We wrote the answer as x2 but why + C ?

It is the "Constant of Integration". It is there because of all the functions whose derivative is 2x:

many integrals vs one derivative

The derivative of x2+4 is 2x, and the derivative of x2+99 is also 2x, and so on! Because the derivative of a constant is zero.

So when we reverse the operation (to find the integral) we only know 2x, but there could have been a constant of any value.

So we wrap up the idea by just writing + C at the end.


Tap and Tank

integral tap tank

Integration is like filling a tank from a tap.

The input (before integration) is the flow rate from the tap.

Integrating the flow (adding up all the little bits of water) gives us the volume of water in the tank.

Simple Example: Constant Flow Rate

integral tap tank constant flow


Integration: With a flow rate of 1, the tank volume increases by x

Derivative: If the tank volume increases by x, then the flow rate is 1

This shows that integrals and derivatives are opposites!


Now For An Increasing Flow Rate

Imagine the flow starts at 0 and gradually increases (maybe a motor is slowly opening the tap).

integral tap tank flow rate 2x gives volume x^2

As the flow rate increases, the tank fills up faster and faster.

Integration: With a flow rate of 2x, the tank volume increases by x2

Derivative: If the tank volume increases by x2, then the flow rate must be 2x

Example: with the flow in liters per minute, and the tank starting at 0

After 3 minutes (x=3):

And after 4 minutes (x=4):


We can do the reverse, too:

integral tap tank 2x

Imagine you don't know the flow rate.
You only know the volume is increasing by x2.

We can go in reverse (using the derivative, which gives us the slope) and find that the flow rate is 2x.


integral vs derivative 

So Integral and Derivative are opposites.

We can write that down this way:

The integral of the flow rate 2x tells us the volume of water:

 2x dx = x2 + C

And the slope of the volume increase x2+C gives us back the flow rate:

 d/dx(x2 + C) = 2x


integral tap tank graphs

And hey, we even get a nice explanation of that "C" value ... maybe the tank already has water in it!

Which teaches us to always add "+ C".

Other functions

Well, we have played with y=2x enough now, so how do we integrate other functions?

If we are lucky enough to find the function on the result side of a derivative, then (knowing that derivatives and integrals are opposites) we have an answer. But remember to add C.

Example: what is cos(x) dx ?

integral vs derivative, cos(x) vs sin(x)

From the Rules of Derivatives table we see the derivative of sin(x) is cos(x) so:

cos(x) dx = sin(x) + C

But a lot of this "reversing" has already been done (see Rules of Integration).

Example: What is x3 dx ?

On Rules of Integration there is a "Power Rule" that says:

xn dx = xn+1n+1 + C

We can use that rule with n=3:

x3 dx = x44 + C

Knowing how to use those rules is the key to being good at Integration.

So get to know those rules and get lots of practice.

Learn the Rules of Integration and Practice! Practice! Practice!
(there are some questions below to get you started)

Definite vs Indefinite Integrals

We have been doing Indefinite Integrals so far.

A Definite Integral has actual values to calculate between (they are put at the bottom and top of the "S"):

indefinite integral definite integral
Indefinite Integral Definite Integral

Read Definite Integrals to learn more.