1. ## true/false help

True (T) or False (F)? Let Int(a,b)(f dx) be the integral from a to b of f with respect to x

_____ Int(1,2)(Int(1,2)((xy dx dy)) = Int(1,2)(Int(1,2)(xy dy dx))
_____ Int(1,2)(Int(1,3)(xy^2) dx dy)) = (Int(1,2)( x dx))(Int(1,3)(y^3) dy))
_____ Int(1,2)(Int(1,2)(xy dx dy)) = [Int(1,2)( x dx)]^2
_____ Int(1,2)(Int(1,x)([f(x,y)^2 dx dy]) = Int(1,x)(Int(1,2)[f(x,y)^2 dy dx for any function f(x,y)
_____ For any mass density the center of mass of a spherical ball is at the center of the spherical ball.

2. Originally Posted by ratedmichael

True (T) or False (F)? Let Int(a,b)(f dx) be the integral from a to b of f with respect to x

_____ Int(1,2)(Int(1,2)((xy dx dy)) = Int(1,2)(Int(1,2)(xy dy dx))
$\displaystyle \int_1^2 \int_1^2 xy~dx~dy = \int_1^2 \int_1^2 xy~dy~dx$
True. The function xy has no "psychotic" behavior, so we can switch the order of integration.

Originally Posted by ratedmichael
_____ Int(1,2)(Int(1,3)(xy^2) dx dy)) = (Int(1,2)( x dx))(Int(1,3)(y^3) dy))
$\displaystyle \int_1^2 \int_1^3 xy^2~dx~dy = \int_1^2 x~dx~\int_1^3 y^2~dy$
False. This would be true if the limits of integration matched on both sides of the equation since the limits of integration have nothing to do with x or y. So we can separate the integrals. But the 1-3 integration limits belong to the x integration, not the y integration.

Originally Posted by ratedmichael
_____ Int(1,2)(Int(1,2)(xy dx dy)) = [Int(1,2)( x dx)]^2
$\displaystyle \int_1^2 \int_1^2 xy~dx~dy = \left ( \int_1^2 x~dx \right )^2$
True. This is an application of the last one. We separate the integrals, then note that the variable of integration of y integral is a "dummy" variable. It goes like this:
$\displaystyle \int_1^2 \int_1^2 xy~dx~dy = \left ( \int_1^2 x~dx \right ) \left ( \int_1^2 y~dy \right )$

$\displaystyle = \left ( \int_1^2 x~dx \right ) \left ( \int_1^2 x~dx \right )$ <-- Replacing the dummy variable y with an x.

$\displaystyle = \left ( \int_1^2 x~dx \right )^2$

Originally Posted by ratedmichael

_____ Int(1,2)(Int(1,x)([f(x,y)^2 dx dy]) = Int(1,x)(Int(1,2)[f(x,y)^2 dy dx for any function f(x,y)
$\displaystyle \int_1^2 \int_1^x f^2(x,y)~dx~dy = \int_1^x \int_1^2 f^2(x,y)~dy~dx$ for any function f(x,y).
I'm going to go with "False" for this one. Again, it's going to depend on how "pathological" the function is in the integration interval. Consider, for example, the function $\displaystyle f(x, y) = \frac{x}{x - y}$. I did it easily doing it dy dx, but even my calculator flatly refused to do the dx dy version. I'm assuming they would come out to be different.

Originally Posted by ratedmichael
_____ For any mass density the center of mass of a spherical ball is at the center of the spherical ball.
False. This one should be easy to see.

-Dan