Let $H = \{(x,y,z): 0 \leq z \leq 2, \frac{1}{2}(1+z^2) \leq x^2+y^2 \leq 2(1+z^2)\}$.
How can one calculate the following intergral?
$$\int_H x^2 z d \lambda_3 (x,y,z)$$
(where $\lambda$ is the Lebesgue integral)
2026-04-03 17:05:13.1775235913
Transformationssatz on balls
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Hint: Change $(x,y)$ to polar coordinates $(r,\theta)$ keeping $z$ unchanged. The limits are $0 \leq \theta \leq 2\pi$, $\frac 1 {\sqrt 2} \sqrt {1+z^{2}} \leq r \leq \sqrt 2 \sqrt {1+z^{2}}$ and $0 \leq z \leq 2$. Integrate w.r.t. $\theta$ then w.r.t. $r$ and finally w.r.t. $z$.