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3.3 Use the differential volume dv to determine the volumes of the following regions: (a) 0

3.3 Use the differential volume dv to determine the volumes of the following regions: (a) 0

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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### Problem 3.3: Using Differential Volume to Determine Volumes of Regions Calculate the volumes of the following regions using the differential volume \( dv \): #### (a) Cartesian Coordinates - **Range for \( x \):** \( 0 < x < 1 \) - **Range for \( y \):** \( 1 < y < 2 \) - **Range for \( z \):** \( -3 < z < 3 \) #### (b) Spherical Coordinates - **Range for \( \rho \):** \( 2 < \rho < 5 \) - **Range for \( \phi \):** \( \pi/3 < \phi < \pi \) - **Range for \( z \):** \( -1 < z < 4 \) #### (c) Cylindrical Coordinates - **Range for \( r \):** \( 1 < r < 3 \) - **Range for \( \theta \):** \( \pi/2 < \theta < 2\pi/3 \) - **Range for \( \phi \):** \( \pi/6 < \phi < \pi/2 \) This problem involves integrating over specified ranges in different coordinate systems to find the volume of the defined regions.
Transcribed Image Text:### Problem 3.3: Using Differential Volume to Determine Volumes of Regions Calculate the volumes of the following regions using the differential volume \( dv \): #### (a) Cartesian Coordinates - **Range for \( x \):** \( 0 < x < 1 \) - **Range for \( y \):** \( 1 < y < 2 \) - **Range for \( z \):** \( -3 < z < 3 \) #### (b) Spherical Coordinates - **Range for \( \rho \):** \( 2 < \rho < 5 \) - **Range for \( \phi \):** \( \pi/3 < \phi < \pi \) - **Range for \( z \):** \( -1 < z < 4 \) #### (c) Cylindrical Coordinates - **Range for \( r \):** \( 1 < r < 3 \) - **Range for \( \theta \):** \( \pi/2 < \theta < 2\pi/3 \) - **Range for \( \phi \):** \( \pi/6 < \phi < \pi/2 \) This problem involves integrating over specified ranges in different coordinate systems to find the volume of the defined regions.
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