1. Integration-Example
Find the area between the 2 curves between the 2 points of intersection. X=1 and X=3
1)The first step in solving an equation like this is a nice clear graph, which will help you visualise the problem. The graph below shows us the 2 curves and their points of intersection but I shall show you how to calculate the points of intersection algebraically.
3) The first step involves finding which is the top curve and bottom curve this can be seen easily from the graph, however if a graph is not present an x value can be substituted in to each equation and the one with the higher y value is the top curve.

2. Area between 2 curves
By calculating Top curve – bottom curve we can calculate an equation which when integrated between the points in question will give the area between the 2 curves.
In this case the top curve is:
And the bottom curve is :

3. Integration
By integrating the curve I will be able to calculate the area between 2 given points.
The sign for integration is and to find the area between the 2 curves between 2 x values we must integrate with respects to x (dx).
To integrate we use this general formula
The square bracket is the notation we use once the curve has been integrated.
This means we add 1 to the power of the x
value then divide this number by the new value
of the power of x.

4. Integrating the curve
These figures are the 2 x values between which I am integrating.
To calculate the area between these 2 points I must (Area from x=3)-(area from x=1) this is because when integrating the x value of 3 it finds the total area between x=3 and x=0 so by subtracting the area found by integrating x=1 from the area when x=3 it will find the area between these 2 points.

5. Continued Integration

6. Testing
We can test if this value is correct by using autograph and the trapezium rule.
Which surprisingly is very precise.