The body K is limited by the cone z= -sqrt(x^2+y^2) and the sphere x^2+y^2+z^2=16

Now, the most natural thing to do is to introduce spherical coordinates:

x= rsinωcosφ

y= rsinωsinφ

z= rcosω

In this case, the body K would be described by:

cone z= -sqrt(x^2+y^2)--> -sqrt(r^2(sinω)^2)= - rsinω

and x^2+y^2+z^2=16 --> r^2= 16

z= rcosω = -rsinω --> ω= -pi/4

which means that the limits of the integral is

0<r<sqrt(16)

-pi/4<ω<0

0<φ<2pi

(is this correct?)

However, the task states that I should use cylindrical coordinates to describe the body K:

x= rcosφ

y= rsinφ

z= z

Then the body K would be described as: the cone = -r and the sphere = r^2 + z=16

But which limits do I set here?

Is this correct, is the transformation to cylindrical spheres this easy when they ask for the body K to be described by cylindrical coordinates?