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.

Integration can be used to find areas, volumes, central points and many useful things. But it is often used to find the area under the graph of a function like this:
The area can be found by adding slices that approach zero in width:And there are Rules of Integration that help us get the answer.

Notation

A mathematical notation is a writing system of symbols used for recording concepts in Mathematics searching for the notational simplicity and meaningfulness. Ideally, it should emphasize the most important features of a mathematical expression or statement, while downplaying the other one.

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).

Definite Integral

A Definite Integral has start and end values: in other words, there is an interval [a, b].

a and b (called limits, bounds or boundaries) are put at the bottom and top of the “S”, like this:

Definite Integral (from a to b)		Indefinite Integral (no specific values)

We find the Definite Integral by calculating the Indefinite Integral at a, and at b, then subtracting:

Notation: We can show the indefinite integral (without the +C) inside square brackets, with the limits a and b after, like this:

Let’s try another example:

And another example to make an important point:

But we can have negative regions, when the curve is below the axis:

So we have this important thing to remember:

Try integrating cos(x) with different start and end values to see for yourself how positives and negatives work.

Positive Area

But sometimes we want all area treated as positive (without the part below the axis being subtracted).

In that case we must calculate the areas separately, like in this example:

Continuous

Oh yes, the function we are integrating must be Continuous between a and b: no holes, jumps or vertical asymptotes (where the function heads up/down towards infinity).

Properties

Area above − area below

The integral adds the area above the axis but subtracts the area below, for a “net value”:

Adding Functions

The integral of f+g equals the integral of f plus the integral of g:

Reversing the interval

Reversing the direction of the interval gives the negative of the original direction.

Interval of zero length

When the interval starts and ends at the same place, the result is zero:

Adding intervals

We can also add two adjacent intervals together:

Area Under a Curve

How to find the area under curves using definite integrals; tutorials, with examples and detailed solutions, are presented. A set of exercises with answers is presented at the bottom of the page. Also, tutorials on the area between curves are included.

VIDEO TUTORIAL