Evaluate the iterated integral:

Evaluate the double integral over the rectangular region $R = [0, 2] \times [1, 2]$:

Evaluate the double integral by changing to polar coordinates, where $D$ is the region bounded by the semi-circle $x = \sqrt{1 - y^2}$ and the y-axis.

Evaluate the triple integral of $f(x,y,z) = xyz$ over the rectangular box $B = [0, 1] \times [0, 2] \times [0, 3]$.

Use cylindrical coordinates to find the volume of the solid $E$ bounded above by the paraboloid $z = 4 - x^2 - y^2$ and below by the xy-plane ($z = 0$).

Use spherical coordinates to evaluate $\iiint_B e^{(x^2+y^2+z^2)^{3/2}} \, dV$, where $B$ is the unit ball $x^2 + y^2 + z^2 \le 1$.

Find the center of mass of a triangular lamina with vertices $(0,0)$, $(1,0)$, and $(0,2)$ if its density function is $\rho(x,y) = 1 + 3x + y$.