Find the area between the curves y = x^2 and y = 4x - x^2.

First, we need to draw a diagram, to understand the question more clearly. 
y = x^2 is a common curve, but to sketch y = 4x - x^2, we should factorise it, to find where it inersects the x-axis.
4x - x^2 = x(4-x). When this equals 0, x = 0 or x = 4, so therse are the points where the curve intersects the x- axis. Now we have the information we need to sketch the curve.
Now we need to find the x- coordiantes of where the curves intersect, which we can see from the diagram is at 2 points. To do this, we equate the equations of the two curves. This gives:
x^2 = 4x - x^2
2x^2 - 4x = 0
x(2x - 4) = 0
So x = 0 or x = 4/2 = 2.
So these will be our limits of integration.
Now we can integrate between the two curves.
We can see from the diagram that the y = 4x - x^2 curve is above the y = x^2  curve in this region, so our integrand will be (4x - x^2) - x^2 = 4x - 2x^2.  So we need to calculate the integral of 4x - 2x^2 between the limits 0 and 2.
This integrates to give 2x^2 - (2/3)x^3. 
Substituting in the limits, we have [2(2)^2 - (2/3)(2)^3] - [2(0)^2 - (2/3)(0)^3] = 8 - 16/3 - 0 + 0 = 8/3.  So our area between the two curves is 8/3.