3. 化三重积分 $\iiint_{\Omega} f(x, y, z) \mathrm{d} x \mathrm{~d} y \mathrm{~d} z$ 为三次积分, 其中积分区域 $\Omega$ 分别是:
(1) 由平面 $y=0, z=0, x+z=\frac{\pi}{2}$ 及拋物柱面 $y=\sqrt{x}$ 所围成的闭区域;
(2) 由圆锥面 $z=2-\sqrt{x^{2}+y^{2}}$ 及抛物面 $z=x^{2}+y^{2}$ 所围成的闭区域;
（3）由双曲抛物面 $z=x y$ 、圆柱面 $x^{2}+y^{2}=1$ 及平面 $z=0$ 所围成的位于第 $\mathrm{I}$ 卦限的闭区域.

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∻Solution by Steps∻ ‖ step 1 ⋮ Identify the integral to be solved: $ \iiint_{\Omega} f(x, y, z) \, dx \, dy \, dz $ ‖ ‖ step 2 ⋮ Determine the region $\Omega$ for each part of the problem: 1. The region bounded by the planes $y=0$, $z=0$, $x+z=\frac{\pi}{2}$, and the parabolic cylinder $y=\sqrt{x}$. 2. The region bounded by the cone $z=2-\sqrt{x^2+y^2}$ and the paraboloid $z=x^2+y^2$. 3. The region bounded by the hyperbolic paraboloid $z=xy$, the cylinder $x^2+y^2=1$, and the plane $z=0$ in the first octant. ‖ ‖ step 3 ⋮ For each region, set up the limits of integration: 1. For the first region: - $0 \leq y \leq \sqrt{x}$ - $0 \leq z \leq \frac{\pi}{2} - x$ - $0 \leq x \leq \frac{\pi}{2}$ 2. For the second region: - $0 \leq z \leq 2 - \sqrt{x^2 + y^2}$ - $0 \leq z \leq x^2 + y^2$ - Convert to cylindrical coordinates: $0 \leq r \leq 1$, $0 \leq heta \leq 2\pi$ 3. For the third region: - $0 \leq z \leq xy$ - $0 \leq x^2 + y^2 \leq 1$ - $0 \leq x \leq 1$, $0 \leq y \leq 1$ ‖ ‖ step 4 ⋮ Evaluate the triple integrals for each region: 1. $\iiint_{\Omega_1} f(x, y, z) \, dx \, dy \, dz = \int_{0}^{\frac{\pi}{2}} \int_{0}^{\sqrt{x}} \int_{0}^{\frac{\pi}{2} - x} f(x, y, z) \, dz \, dy \, dx$ 2. $\iiint_{\Omega_2} f(x, y, z) \, dx \, dy \, dz = \int_{0}^{2\pi} \int_{0}^{1} \int_{r^2}^{2-r} f(r \cos heta, r \sin heta, z) \, r \, dz \, dr \, d heta$ 3. $\iiint_{\Omega_3} f(x, y, z) \, dx \, dy \, dz = \int_{0}^{1} \int_{0}^{1} \int_{0}^{xy} f(x, y, z) \, dz \, dy \, dx$ ‖ ∻Answer∻ ⚹ The triple integrals for the given regions are set up as shown in step 4. ⚹ ∻Key Concept∻ ⚹Triple Integral Setup⚹ ∻Explanation∻ ⚹The key concept is setting up the limits of integration for the given regions and converting to appropriate coordinates if necessary.⚹◊What is the meaning of the triple integral $\iiint_{\Omega} f(x, y, z) \mathrm{d} x \mathrm{d} y \mathrm{d} z$ in the context of the information provided by the OCR model⍭ Generate me a similar question◊

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