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Evaluate the integral S (10 – 2z) dS, where S is the part of the surface z = S inside the cylinder x2 + y² = 1. 2 2

Evaluate the integral S (10 – 2z) dS, where S is the part of the surface z = S inside the cylinder x2 + y² = 1. 2 2

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 Statement:** Evaluate the integral \(\iint_S (10 - 2z) \, dS\), where \(S\) is the part of the surface described by the equation \(z = 5 - \frac{x^2}{2} - \frac{y^2}{2}\) that lies inside the cylinder \(x^2 + y^2 = 1\). **Explanation:** The integral is taken over a surface \(S\) that is defined by a quadratic equation in terms of \(x\) and \(y\). This surface represents a paraboloid opening downward, with its vertex at \((0, 0, 5)\). The region of integration, \(x^2 + y^2 = 1\), defines a circular boundary (a cylinder) in the \(xy\)-plane with a radius of 1 centered at the origin. The integral \(\iint_S (10 - 2z) \, dS\) involves evaluating the expression \(10 - 2z\) over the surface \(S\) using the surface differential element \(dS\). This type of problem often involves converting the surface integral into a more manageable form by using parameters that describe the surface or simplifying using symmetry and geometry considerations.
Transcribed Image Text:**Problem Statement:** Evaluate the integral \(\iint_S (10 - 2z) \, dS\), where \(S\) is the part of the surface described by the equation \(z = 5 - \frac{x^2}{2} - \frac{y^2}{2}\) that lies inside the cylinder \(x^2 + y^2 = 1\). **Explanation:** The integral is taken over a surface \(S\) that is defined by a quadratic equation in terms of \(x\) and \(y\). This surface represents a paraboloid opening downward, with its vertex at \((0, 0, 5)\). The region of integration, \(x^2 + y^2 = 1\), defines a circular boundary (a cylinder) in the \(xy\)-plane with a radius of 1 centered at the origin. The integral \(\iint_S (10 - 2z) \, dS\) involves evaluating the expression \(10 - 2z\) over the surface \(S\) using the surface differential element \(dS\). This type of problem often involves converting the surface integral into a more manageable form by using parameters that describe the surface or simplifying using symmetry and geometry considerations.
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