
Double Integration
I am just learning about double and triple integration, but I am stuck on this problem:
Find the average distance to the xaxis for points in the region bounded by the xaxis and the graph $\displaystyle y = x  x^2$
I am not quite sure how to set up the double integral. I appreciate the help.

The "distance to the xaxis" for any point (x, y) is, of course, y.
To find the average value of function f(x,y) over twodimensional 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 xaxis and the graph "y= x x^2[/tex]". If draw a graph you will see that that graph is a parabola that intersects the xaxis 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 $\displaystyle \frac{\int_{x=0}^1\int_{y= 0}^{x x^2} y dy dx}{\int_{x=0}^1\int_{y=0}^{x x^2} dy dx}$.