# Double Integration

• Apr 22nd 2010, 06:07 AM
Double Integration
I am just learning about double and triple integration, but I am stuck on this problem:

Find the average distance to the x-axis for points in the region bounded by the x-axis and the graph $\displaystyle y = x - x^2$

I am not quite sure how to set up the double integral. I appreciate the help.
• Apr 22nd 2010, 09:09 AM
HallsofIvy
The "distance to the x-axis" for any point (x, y) is, of course, y.

To find the average value of function f(x,y) over two-dimensional region U, take the integral of f(x,y) over that region, divided by the area of the region:
$\displaystyle \frac{\int_U f(x,y) dxdy}{\int_U dxdy}$.

Here, you region is "bounded by the x-axis and the graph "y= x- x^2[/tex]". If draw a graph you will see that that graph is a parabola that intersects the x-axis at x= 0 and x= 1. In order to cover that region, you will have to take x going from 0 to 1 and, [b]for each x, y from 0 to $\displaystyle x- x^2$.

You want $\displaystyle \frac{\int_{x=0}^1\int_{y= 0}^{x- x^2} y dy dx}{\int_{x=0}^1\int_{y=0}^{x- x^2} dy dx}$.