Area Bounded Partly By A Curve

Recap
We already know that evaluating the integral found by:
Gives the area bound by x=a, x=b, y=f(x) and the x-axis, e.g.

Areas below the x-axis

Areas below the x-axis

An area is always positive.
The definite integral is positive for areas above the x-axis but negative for areas below the axis.
To find an area, we need to know whether the curve crosses the x-axis between the boundaries.
For areas above the axis, the definite integral gives the area.
For areas below the axis, we need to change the sign of the definite integral to find the area.
Areas below the x-axis

Example
Evaluate:
We can see that the function has 3 roots at:
x = -1, x = 0 and x = 2
So the function does cross the x-axis between the limits at x = 0.
Therefore we consider the question in two parts:

Example
Our knowledge of a positive cubic function tells us that the section between the limits -1 and 0 is above the x-axis and the section between the limits 0 and 2 is below the x-axis:
So the first integral is positive but the second will give a negative result and so should be multiplied by -1.

Example
So the total area is given by:

Areas bound with the y-axis
We can also evaluate areas bound by y=a, y=b, x=f(y) and the y-axis by evaluating:
An example would look like this:

Example
Find this area bound by y=1, y=3, y=√(x-2) and the y-axis.
Firstly we need to make x the subject of the equation: x = y2 + 2
Now we have to evaluate: