I don't understand where is my mistake on calculating the volume by the second way.
The volume that I want to calculate is bounded by $z = 5$ and $z^2=x^2+y^2$, so it is the upper part of the cone, simple example.
The first way is to define $z$ as a constant $0 \leq z\leq 5$ and then do the substitution $x = r \cos(\phi)$, $y = r \sin(\phi)$, when $0 \leq r \leq z$ and $0\leq \phi \leq 2\pi$. That way I get the correct answer.
The second way I can choose D as $x^2+y^2 \leq 25$ and I want to do $$\int \int dx dy $$ on $$\int 1dz$$ when $D = {(x,y) = x^2 + y^2 = 25}$ and the $\int 1dz$ is between 5 and $\sqrt{x^2+y^2}$. So evantually I get the integral $$\int \int \left(5 - \sqrt{x^2+y^2}\right) dxdy $$
The last method called in our terminology "stick method" because we took D in 2D, and pulling sticks from the D to the surface that bounds it. And the first method called "Slice method" because here we do slices and calculating the sum of them from constant to constant (it can be z, x or y that bounded by 2 numbers). I hope I explained myself brightly. Can somebody correct the second method and tell me what am I doing wrong in it ?
P.S - in the second way I use the trigonometrical substitution too (after I write it in the end).