An integral is a way of adding slices to find a whole. An integral is an operation used to measure or calculate the area under a curve of a mathematical function, which is often referred to as an “integral”.
Slice
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):
We can make Δx a lot smaller and add up many small slices (answer is getting better):
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.
But that’s too many slices to add, but here we don’t need to add them, there is a shortcut, because finding an Integral is the opposite of finding a Derivative.
Example:
An integral of 2x is
because the derivative of is
That simple example can be confirmed by calculating the area:
Area of triangle =
Notation
The symbol for “Integral” is a stylish “S” (for “Sum”, the idea of summing slices):
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:
Plus C
We wrote the answer as but why ?
It is the “Constant of Integration”. It is there because of all the functions whose derivative is :
- The derivative of is ,
- and the derivative of is also ,
- and the derivative of is also ,
- and so on!
Because the derivative of a constant is zero.
So when we reverse the operation (to find the integral) we only know , but there could have been a constant of any value.
So we wrap up the idea by just writing at the end.
A Practical Example: Tap and Tank
Let us use a tap to fill a tank.
The input (before integration) is the flow rate from the tap.
We can integrate that flow (add up all the little bits of water) to give us the volume of water in the tank.
Imagine a Constant Flow Rate of 1:
With a flow rate of 1, the tank volume increases by x. That is Integration!
An integral of 1 is x
With a flow rate of 1 liter per second, the volume increases by 1 liter every second, so would increase by 10 liters after 10 seconds, 60 liters after 60 seconds, etc.
The flow rate stays at 1, and the volume increases by x
And it works the other way too:
If the tank volume increases by x, then the flow rate must be 1.
The derivative of x is 1
This shows that integrals and derivatives are opposites!
