Calculate the volume $λ_3(A)$ of $A := \{ (x, y, z) ∈\mathbb{R}^3 : x^2 ≤ y^2 + z^2 ≤ 1\} $ I have a lot of problems with exercises where I must calculate the Volume of a set using integrals.
Here an example:
Le the set $A$ be $A := \{ (x, y, z) ∈\mathbb{R}^3 : x^2 ≤ y^2 + z^2 ≤ 1\} $. Calculate its volume $λ_3(A)$.
So what I thought is to doing something like this:
$λ_3(A)=λ_3(A_1)-λ_3(A_2)$
Where $A_1 := \{ (x, y, z) ∈\mathbb{R}^3 : y^2 + z^2 ≤ 1\}$ and $A_2 := \{ (x, y, z) ∈\mathbb{R}^3 : x^2 ≤ y^2 + z^2 \}$
But here comes my problem for every exercise like this: I dont know how to find the limits of my triple integral.
Can someone help me?

$\iiint_Adxdydz=\int_{-1}^1C(x)dx$ where $C(x)=\iint_{x^2\le y^2+z^2\le1}dydz=\pi(1-x^2)$ (area of an annulus).
Hence
$$\lambda_3(A)=\pi\int_{-1}^1(1-x^2)dx=\pi\left[x-\frac{x^3}3\right]_{-1}^1=\frac{4\pi}3.$$
