# Using integration to find an area

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## Using integration to find an area

The fact that **integration** can be used to find the area under a graph comes from the idea of splitting the graph into small 'rectangles' and adding up their areas. **It works as follows:**

The area is the sum of all the heights (the y-values) multiplied by the width (δx) or Σyδx

As we allow δx → 0 this approaches the area =

In order to find the area under a graph we need to state the lower and upper values of x. These values are called the **limits** and are written at the top and bottom of the integral sign to indicate which area is being found.

**Example:**

To find this area we have to integrate between x = 1 and x = 4. This is written as:

**To evaluate this we integrate and then substitute in the limits, subtracting the value of the lower limit from the value of the higher limit.**

Note that we have ignored the '+ c'. *(Try including it and see what happens.) *

When using this method to find an area below the x-axis the final value will be negative. So when giving a final answer remember that an area has to be positive − the minus sign just means the area is below the x-axis.

**This can give rise to problems when the limits include parts of the graph above and below the axis. In order to cope with this we need to:**

- Sketch the graph.
- Split the graph into zones where the area is only above or below the axis.
- Integrate each of these zones separately.
- Add up the separate areas at the end, (remembering to ignore the minus signs).

**Example:**

Find the total area between the graph x^{3} - x^{2} - 2x and the x axis, between the points x = -1 and x = 2.

The area asked for above can be using

but you must be careful as the graph has negative values of x in the region mentioned. Follow the 4 steps listed above...

1. **Sketch the graph**

In order to sketch the graph we need to factorise to help us find where the graph crosses the x-axis.

x^{3} − x^{3} − 2x = x(x^{2} − x − 2) = x(x - 2)(x + 1) = 0,
when x = 0, x = 2, or x = -1.

This means the graph looks like this:

2. **Split the graph into zones where the area is only above or below the axis.**

From the graph we can see that the graph is above the x-axis for -1 < x < 0, and below the x-axis for 0 < x < 2. Therefore we will integrate in these two zones.

3. **Integrate each of these zones separately.**

4. **Add up the separate areas at the end, (remembering to ignore the minus signs).**

The total area between the graph and the x-axis =

This uses the same principles as before, except we have effectively swapped the axes over.

This means we have to find

where the limits are y-values and the equation of the line is rewritten as x = f(y).

**Example:**

Find the area between the curve y = √(x − 2), the y-axis and the lines y = 1, and y = 3.

Rearrange the equation to get x = y^{2} + 2, and then integrate this between the limits y = 1 and y = 3 to get: