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:

Rectangles

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 =

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.

Limits

Example:

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

Example

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.

Example

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:

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

Example:

Find the total area between the graph x3 - x2 - 2x and the x axis, between the points x = -1 and x = 2.

The area asked for above can be using

Example

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.

x3 − x3 − 2x = x(x2 − x − 2) = x(x - 2)(x + 1) = 0, when x = 0, x = 2, or x = -1.

This means the graph looks like this:

Graph

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.

Example

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 =

Example

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

This means we have to find

Area between the curve and the y-axis

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.

Example

Rearrange the equation to get x = y2 + 2, and then integrate this between the limits y = 1 and y = 3 to get:

Example

S-cool exclusive!!