## Integration: the area under a curve

### Definite Integration

When we work out a definite integral we end up with a value as opposed to a function

A definite integral is in the form

To calculate

We integrate (find the inverse derivative)

We substitute in into this new function

Then substitute the second value from the first

#### Notation

### Area under a curve

To find the area under a curve we find the definite integral between the two bounds (ends)

#### Proof

Examples

Find the area (A) under the curve and the lines

This is to be expected as all we had was rectangle that was 4 high and 3 wide

Find the area (A) under the curve and the lines(note: we do not require the constant of integration because it will be cancelled out in the subtraction)

Find the area (A) under the curve and the lines

### Areas below the x axis

Let’s look at an example

Here, we have the same rectangle as in the first example, but this time it is below the x axis. The result is that the area is given as negative.

We would report the area as positive, but understand that it is below the axis.

Example

Find the area (A) between the x axis the lines and the curve

### When areas are above and below the axis

Let’s consider the line and go from

If we look at the graph we can see that there is an area, so what has happened.

Well the area below the axis has cancelled out the area above.

To calculate the desired area we must find the area from 0 to 2 and then from 2 to 4.

So has an area of 4 (area is positive)

So has an area of 4

Examples

Find the area between the x axis, and the curve

First we find if the function equals zero between

let

Since it crosses the axis at we need to integrate from 0 to 1 and then from 1 to 2. We will call these areas .

Since

So the area required

Find the area between the x axis, and the curveFirst we find if the function equals zero between

let

Since it crosses the axis at we need to integrate from 0 to 1 and then from 1 to 4. We will call these areas .

So the area required

## Integration, the inverse of differentiation

Integration can be used to reverse the effect of differentiation.

The rule for differentiation is: if then

So we multiply by the power and then take one off the power.

If we reverse this process we get add one to the power and divide by this new power.

So if then

Lets consider what happens when we differentiate a constant i.e.

This is the same as

So if we differentiate a constant we get zero which means if we integrate zero we get a constant.

So then

Examples

, what is a general solution for y?

To get y we need to integrate

So

This is a general solution because could have any value.

Here I have given values that range for -1 to 3, but there were an infinite other possibilities. The important thing to note is that if we differentiate any of these curves they all give the same expression.

The gradient function of a curve is , the curve passes through the point .

What is the particular solution.

Integrating gives

Since the curve goes through we can substitute in these values to get

This gives the particular solution

A particular solution is one where the value of c has been found so that now we can only draw one particular curve.

### Notation

means integrate . The means that the variable we are considering is x.

Example

Find the general solution of