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A) Determine the boundary limits of the following regions in spaces. The region D₁ bounded by the planes x + 2y + 3z = 6 and the coordinate planes. The region D₂ bounded by the cylinders y = x² and y = 4 - x², and the planes x + 2y + z = 1 and x + y + z = 1. The region D3 bounded by the cone z² = x² + y² and the parabola z = 2-x² - y² The region D4 in the first octant bounded by the cylinder - x² + y² = 4, the paraboloid z = 8 - x2 - y² and the planes x = y, z = 0, and x = 0. B) Calculate the following integrals JJJ. dV, y dV JJJ.* xy dV, [11. D4 dV,

A) Determine the boundary limits of the following regions in spaces. The region D₁ bounded by the planes x + 2y + 3z = 6 and the coordinate planes. The region D₂ bounded by the cylinders y = x² and y = 4 - x², and the planes x + 2y + z = 1 and x + y + z = 1. The region D3 bounded by the cone z² = x² + y² and the parabola z = 2-x² - y² The region D4 in the first octant bounded by the cylinder - x² + y² = 4, the paraboloid z = 8 - x2 - y² and the planes x = y, z = 0, and x = 0. B) Calculate the following integrals JJJ. dV, y dV JJJ.* xy dV, [11. D4 dV,

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Just calculate the integral of region D3 and D4 ( B part). [The boundary limits of D3 and D4 are already given in the image below]
A) Determine the boundary limits of the following regions in spaces. The region D₁ bounded by the planes x + 2y + 3z: 6 and the coordinate planes. The region D2₂ bounded by the cylinders y = x² and y = 4 - x², and the planes x + 2y + z = 1 and x + y + z = 1. The region D3 bounded by the cone z² = x² + y2 and the parabola z=2-x² - y² The region D4 in the first octant bounded by the cylinder x² + y² = 4, the paraboloid z = 8 - x² - y² and the planes x = y, z = 0, and x = = 0. B) Calculate the following integrals JJJ. dV, III yov [][ J. xy dV JJJ. dV₁
Transcribed Image Text:A) Determine the boundary limits of the following regions in spaces. The region D₁ bounded by the planes x + 2y + 3z: 6 and the coordinate planes. The region D2₂ bounded by the cylinders y = x² and y = 4 - x², and the planes x + 2y + z = 1 and x + y + z = 1. The region D3 bounded by the cone z² = x² + y2 and the parabola z=2-x² - y² The region D4 in the first octant bounded by the cylinder x² + y² = 4, the paraboloid z = 8 - x² - y² and the planes x = y, z = 0, and x = = 0. B) Calculate the following integrals JJJ. dV, III yov [][ J. xy dV JJJ. dV₁
18:49 G ← Step2 b) (3 Here (4) => Z² So 22= x2+y2 = x² + y² = (2-be²+ y²)) ² y varies from x varies from Putting x=y y 2 u = (2-4)² 4 = 4-44+4² 4²-$4+4=0 (4-4) (4-1)=0 4=1,4 x² + y² = 1₂4 but from 2 = 2-(x² + y²), x² + y² ≤ 2 Hence x² + y² = 1 (This gives the limits for x,y) Here z varies from √₂z+yz to 2-x²-y2. √1-x² to √1-x2 -1 1 ||| in 2 =2-x²-y² O (Let u= x² + y²) х2+у? 4 x²+x²=4 2x²=4 z varies from 0 to 8-x²-y² x varies from y to √4-yz (from cylinded varies from to √2 31% x² = 2 = x = √₂ ⇒ y = √2 √x < Do
Transcribed Image Text:18:49 G ← Step2 b) (3 Here (4) => Z² So 22= x2+y2 = x² + y² = (2-be²+ y²)) ² y varies from x varies from Putting x=y y 2 u = (2-4)² 4 = 4-44+4² 4²-$4+4=0 (4-4) (4-1)=0 4=1,4 x² + y² = 1₂4 but from 2 = 2-(x² + y²), x² + y² ≤ 2 Hence x² + y² = 1 (This gives the limits for x,y) Here z varies from √₂z+yz to 2-x²-y2. √1-x² to √1-x2 -1 1 ||| in 2 =2-x²-y² O (Let u= x² + y²) х2+у? 4 x²+x²=4 2x²=4 z varies from 0 to 8-x²-y² x varies from y to √4-yz (from cylinded varies from to √2 31% x² = 2 = x = √₂ ⇒ y = √2 √x < Do
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