On surface of $C = \{ x^2+y^2=1,0

On the surface of the cylinder $C = \{ x^2 + y^2 = 1, \ 0<z<1 \}$ there is subset A that is measurable in terms of $\lambda_2$.

Additionally, we know that: $A_t = A \cap \{z <t \} , 0<t<1$

Prove that:

$$\int\limits_0^1 \lambda_2(A_t) dt = \iint\limits_{A} (1-z) d \lambda_2$$

I know that such cyliner can be parametrized with cylindrical coordinates, those are:

• $x = r \cos(\theta) = \cos(\theta)$
• $y = r \sin(\theta) = \sin(\theta)$
• $z= z$

for: $\theta \in (0, 2\pi)$ and $r = 1$

But it doesn't help in any way with my inequality. I don't know how to use those subsets of A that appear on the LHS. Any help would be much appreciated.