1. **Problem 44:** Find the solid cylinder that lies on or below the plane $z=8$ and on or above the disk in the $xy$-plane with center the origin and radius 2. 2. **Step 1:** State the problem clearly. We want the solid region bounded above by the plane $z=8$ and below by the disk $x^2 + y^2 \leq 4$ in the $xy$-plane (where $z=0$). 3. **Step 2:** Write the inequalities describing the solid. The disk in the $xy$-plane is given by: $$x^2 + y^2 \leq 2^2 = 4$$ The solid lies between $z=0$ (the disk plane) and $z=8$ (the plane above), so: $$0 \leq z \leq 8$$ 4. **Step 3:** Combine the inequalities to describe the solid cylinder: $$\{(x,y,z) \mid x^2 + y^2 \leq 4, 0 \leq z \leq 8\}$$ 5. **Problem 45:** Find the region consisting of all points between (but not on) the spheres of radius $r$ and $R$ centered at the origin, where $r < R$. 6. **Step 1:** State the problem clearly. We want the set of points strictly between two concentric spheres of radii $r$ and $R$. 7. **Step 2:** Write the inequalities for the spheres. The sphere of radius $r$ is: $$x^2 + y^2 + z^2 = r^2$$ The sphere of radius $R$ is: $$x^2 + y^2 + z^2 = R^2$$ 8. **Step 3:** The region between (but not on) the spheres is: $$r^2 < x^2 + y^2 + z^2 < R^2$$ 9. **Problem 46:** Find the solid upper hemisphere of the sphere of radius 2 centered at the origin. 10. **Step 1:** State the problem clearly. We want the upper half of the sphere $x^2 + y^2 + z^2 = 4$ where $z \geq 0$. 11. **Step 2:** Write the equation of the sphere: $$x^2 + y^2 + z^2 = 2^2 = 4$$ 12. **Step 3:** The upper hemisphere is the set of points satisfying: $$x^2 + y^2 + z^2 \leq 4 \quad \text{and} \quad z \geq 0$$ **Final answers:** - Problem 44: $$\{(x,y,z) \mid x^2 + y^2 \leq 4, 0 \leq z \leq 8\}$$ - Problem 45: $$\{(x,y,z) \mid r^2 < x^2 + y^2 + z^2 < R^2\}$$ - Problem 46: $$\{(x,y,z) \mid x^2 + y^2 + z^2 \leq 4, z \geq 0\}$$