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1076893826's Question
Posted 2 months ago

3. 化三重积分 Ωf(x,y,z)dx dy dz\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 Ω\Omega 分别是:
(1) 由平面 y=0,z=0,x+z=π2y=0, z=0, x+z=\frac{\pi}{2} 及拋物柱面 y=xy=\sqrt{x} 所围成的闭区域;
(2) 由圆锥面 z=2x2+y2z=2-\sqrt{x^{2}+y^{2}} 及抛物面 z=x2+y2z=x^{2}+y^{2} 所围成的闭区域;
(3)由双曲抛物面 z=xyz=x y 、圆柱面 x2+y2=1x^{2}+y^{2}=1 及平面 z=0z=0 所围成的位于第 I\mathrm{I} 卦限的闭区域.
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Answer from Sia
Posted 2 months ago
Solution by Steps
step 1
Identify the integral to be solved: Ωf(x,y,z)dxdydz \iiint_{\Omega} f(x, y, z) \, dx \, dy \, dz
step 2
Determine the region Ω\Omega for each part of the problem: 1. The region bounded by the planes y=0y=0, z=0z=0, x+z=π2x+z=\frac{\pi}{2}, and the parabolic cylinder y=xy=\sqrt{x}. 2. The region bounded by the cone z=2x2+y2z=2-\sqrt{x^2+y^2} and the paraboloid z=x2+y2z=x^2+y^2. 3. The region bounded by the hyperbolic paraboloid z=xyz=xy, the cylinder x2+y2=1x^2+y^2=1, and the plane z=0z=0 in the first octant
step 3
For each region, set up the limits of integration: 1. For the first region: - 0yx0 \leq y \leq \sqrt{x} - 0zπ2x0 \leq z \leq \frac{\pi}{2} - x - 0xπ20 \leq x \leq \frac{\pi}{2} 2. For the second region: - 0z2x2+y20 \leq z \leq 2 - \sqrt{x^2 + y^2} - 0zx2+y20 \leq z \leq x^2 + y^2 - Convert to cylindrical coordinates: 0r10 \leq r \leq 1, 0θ2π0 \leq \theta \leq 2\pi 3. For the third region: - 0zxy0 \leq z \leq xy - 0x2+y210 \leq x^2 + y^2 \leq 1 - 0x10 \leq x \leq 1, 0y10 \leq y \leq 1
step 4
Evaluate the triple integrals for each region: 1. Ω1f(x,y,z)dxdydz=0π20x0π2xf(x,y,z)dzdydx\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. Ω2f(x,y,z)dxdydz=02π01r22rf(rcosθ,rsinθ,z)rdzdrdθ\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 \theta, r \sin \theta, z) \, r \, dz \, dr \, d\theta 3. Ω3f(x,y,z)dxdydz=01010xyf(x,y,z)dzdydx\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
The triple integrals for the given regions are set up as shown in step 4.
Key Concept
Triple Integral Setup
The key concept is setting up the limits of integration for the given regions and converting to appropriate coordinates if necessary.

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