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Use "Polar" coordinates to find the "Volume" of the given solid. a) What is the maximum bound of the radius? b) Express the Integration Domain. c) Write the equation for the Volume in (x,y) coordinates d) Perform Volume integration in polar coordinates Bounded by the paraboloids (z = 5 - 2x² and (4x² + 4y² = z) - 2y²)

Use "Polar" coordinates to find the "Volume" of the given solid. a) What is the maximum bound of the radius? b) Express the Integration Domain. c) Write the equation for the Volume in (x,y) coordinates d) Perform Volume integration in polar coordinates Bounded by the paraboloids (z = 5 - 2x² and (4x² + 4y² = z) - 2y²)

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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**Using Polar Coordinates to Find the Volume of a Solid** We are tasked with finding the volume of a given solid using polar coordinates. The solid is bounded by the paraboloids defined by the equations \( z = 5 - 2x^2 - 2y^2 \) and \( 4x^2 + 4y^2 = z \). **Steps to Find the Volume:** a) **What is the maximum bound of the radius?** To determine the maximum bound for the radius \( r \) in polar coordinates, set the given paraboloid equations equal to each other and solve for \( r \) in terms of polar coordinates where \( x = r\cos\theta \) and \( y = r\sin\theta \). b) **Express the Integration Domain.** Express the region of integration in polar coordinates. Determine the limits for \( r \) and \( \theta \) based on the intersection of the paraboloids. c) **Write the Equation for the Volume in (x, y) Coordinates.** Formulate the equation representing the volume based on the difference between the upper and lower surface equations within the integration domain. d) **Perform Volume Integration in Polar Coordinates.** Integrate the volume expression using polar coordinates by converting \( x \) and \( y \) to \( r\cos\theta \) and \( r\sin\theta \), respectively. Evaluate the integral over the defined bounds of \( r \) and \( \theta \). This approach involves accurate setup and execution of integration limits and the conversion of Cartesian coordinates to polar coordinates. The problem underscores the utilization of polar coordinates in calculating volumes of rotationally symmetric solids.
Transcribed Image Text:**Using Polar Coordinates to Find the Volume of a Solid** We are tasked with finding the volume of a given solid using polar coordinates. The solid is bounded by the paraboloids defined by the equations \( z = 5 - 2x^2 - 2y^2 \) and \( 4x^2 + 4y^2 = z \). **Steps to Find the Volume:** a) **What is the maximum bound of the radius?** To determine the maximum bound for the radius \( r \) in polar coordinates, set the given paraboloid equations equal to each other and solve for \( r \) in terms of polar coordinates where \( x = r\cos\theta \) and \( y = r\sin\theta \). b) **Express the Integration Domain.** Express the region of integration in polar coordinates. Determine the limits for \( r \) and \( \theta \) based on the intersection of the paraboloids. c) **Write the Equation for the Volume in (x, y) Coordinates.** Formulate the equation representing the volume based on the difference between the upper and lower surface equations within the integration domain. d) **Perform Volume Integration in Polar Coordinates.** Integrate the volume expression using polar coordinates by converting \( x \) and \( y \) to \( r\cos\theta \) and \( r\sin\theta \), respectively. Evaluate the integral over the defined bounds of \( r \) and \( \theta \). This approach involves accurate setup and execution of integration limits and the conversion of Cartesian coordinates to polar coordinates. The problem underscores the utilization of polar coordinates in calculating volumes of rotationally symmetric solids.
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