## Integration

## Definite Integration

When given

**two x co-ordinates**(limits) you can use definite integration to give the**area between**the**line**and the**x axis**The

This integrates to:

**5**and**2**are the**x limits**This integrates to:

Substitute in the x values giving:

y = 17.5 + c and y = 4 + c

To find the area subtract the larger area from the smaller one:

(17.5 + c) - (4 + c) = 13.5

Generally give the result as fractions so to give the

y = 17.5 + c and y = 4 + c

To find the area subtract the larger area from the smaller one:

(17.5 + c) - (4 + c) = 13.5

**('c's cancel out)**Generally give the result as fractions so to give the

**exact**area
There is a
TRAP!Some integrals may go
below the x axis. When this occurs the integral produces a negative result. This is a problem as this means that instead of adding the two areas from either side of the x axis, you are subtracting one from another. To avoid this draw out the graph. Find out the x intercept. Find out the integrals on either side and add them together to find the area. |

You also need to be able to find the area between a

The

Find the

**curve**and a**straight line**.The

**limits**are where the**two lines meet**. Find this by putting the two equations equal to each other.Find the

**two areas**using integration between the two limits (the trap still affects this result) and then minus one area from the other depending on which graph goes furthest away from the x axis within the limits## Trapezium Rule

**h = width of a strip**

y0 = first y value

yn = last value

y0 = first y value

yn = last value

The Trapezium Rule takes a strip of a certain width and assumes that the two points it meets the line forms a trapezium. This rule then works out the sum of the areas of these trapeziums. This is why there is the '≈' as this means that this is an approximate area. If you do the integral of the line it would show a similarity to the trapezium rule but they wouldn't be the same values. The thinner the strips are the closer the trapezium rule gets to the actual area.

In an exam a table of values will be given that can be used to find out the parts of the equation.

**This is on the equation sheet**