HELP PLEASE!!!! find the area of the region bounded by y=4x-x^2 and y=0

Question

e.mccormick · Answer

The first thing to realize is what area you are dealing with.  y=0 is the x axis, so it is between something and the x axis.  $y=4x-x^2$ is a parabola, but more importantly the square part is negative so it is facing down.  This means you have some sort of parabolic arch.

Next, you need to find the points of intersection.  This defines where the area starts and stops.  They both =y, so you can set one equation equal to the other.  This gets you  $0=4x-x^2$.  Once you solve that you will have the bounds of the integral used to solve this.  If I factor out an x, I get $0=x(4-x)$ so the intersections are at 0 and 4.  

Now I can do a quick graph just to get an idea of what I am talking about.  An arch over the x with intersections at 0 and 4.|dw:1365865921437:dw|
This is not to scale, it is just to get the concept.