Question

Please help me with this calculus question!

Answer 1

the first equation is a sphere centered at (0,0,1/2) with radius 1/2.
to see this, complete the square on the z variable
\[ x^2 + y^2 + z^2 =z \\ x^2 + y^2 + z^2-z +(1/4) = 1/4 \\
x^2+ y^2 + (z-\frac{1}{2})^2 = \frac{1}{4} \]

in cylindrical coords,
\[ r^2 + \left(z-\frac{1}{2}\right)^2 = \frac{1}{4} \\
r= \sqrt{ \frac{1}{4} -\left(z-\frac{1}{2}\right)^2 }
\]

Answer 2

the second equation is a cone with its apex at the origin
\[ z= \frac{1}{\sqrt{3} }\sqrt{x^2+y^2} \]
in cylindrical coords
\[ z = \frac{r}{\sqrt{3} } \\ r = \sqrt{3} \ z
\]

Answer 3

the composite shape is similar to an ice-cream cone: