1.Find the area, mass, and center of mass of a thin plate bounded by $\displaystyle x^2-y^2 =1 $ and $\displaystyle x = 4 $ with density $\displaystyle \rho(x,y) = x $.

Computing these, should be OK. Just wondering about the limits of integration. Would it be $\displaystyle x = 0 $ to $\displaystyle x = 4 $ and $\displaystyle y = 0 $ to $\displaystyle y = \sqrt{x^2-1} $?

2.Let $\displaystyle f(x,y) = \begin{cases} 1, \ \ \ \ \ \text{if} \ x \ \text{is irrational} \\ 4y^{3}, \ \ \text{if not} \end{cases} $

Show that $\displaystyle \int_{0}^{1} \left(\int_{0}^{1} f(x,y) \ dy \right) \ dx = 1 $, but that $\displaystyle \int_{0}^{1} \left(\int_{0}^{1} f(x,y) \ dx \right) \ dy $ does not exist.

Then this implies that $\displaystyle \int_{0}^{1} f(x,y) \ dy = 1 $ or $\displaystyle \int f(x,y) \ dy = y $. So we only integrate it when $\displaystyle f(x,y) = 1 $? And for the second case, the function may be similar to $\displaystyle \int_{0}^{1} \frac{1}{x^2} \ dx $ which does not exist?

The only problem is, when we are integrating from $\displaystyle 0 $ to $\displaystyle 1 $, we integrate over both rationals and irrationals. So maybe the second case deals with this, and that is why it doesn't exist. Whereas, in the first case, we keep $\displaystyle x $ constant, and integrate over the rationals between $\displaystyle 0 $ and $\displaystyle 1 $?