Final answer:
To determine the limits of integration for a half cylinder with a length of 2 and a radius of 2, we consider the half-cylinder oriented along the x-axis which sets the limits from -1 to 1 or 0 to 2, and integrate over the radial distance from 0 to 2 and the angular range from 0 to π.
Step-by-step explanation:
To write the limits of integration for the integral where the region of integration is a half cylinder with length 2 and radius 2, we'll consider various cross-sections and orientations.
For the length along the x-axis, given that the half-cylinder has a length of 2, the limits would be from x = -1 to x = 1 if the axis is in the middle of the rod, or from x = 0 to x = 2 if the axis is at one end of the rod.
For the radial part along the y-axis, since we have a half-cylinder, the limits would range from y = 0 to y = 2, representing the radius of the cylinder.
Finally, for the angular part along the z-axis, considering that it's a half-cylinder, the limits would range from θ = 0 to θ = π, representing half of the full circular cross-section.
The chosen coordinates of x, y, and θ are in a cylindrical coordinate system with the origin at the center of the base of the cylinder, as typically used for cylindrical symmetry.