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Find the centroid of the portion of the sphere x² + y² + z² = 1 with x ≥ 0, y ≥ 0, and z ≥ 0. (Assume uniform density.)

Find the centroid of the portion of the sphere x² + y² + z² = 1 with x ≥ 0, y ≥ 0, and z ≥ 0. (Assume uniform density.)

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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**4.** Find the centroid of the portion of the sphere \( x^2 + y^2 + z^2 = 1 \) with \( x \geq 0 \), \( y \geq 0 \), and \( z \geq 0 \). (Assume uniform density.) **5.** Let \( S \) be the surface in the shape of the parabolic bowl \( z = x^2 + y^2 \) with \( z \leq 1 \). Find the flux of \( \mathbf{F} = x \mathbf{i} + y \mathbf{j} \) across \( S \) in the downward direction.
Transcribed Image Text:**4.** Find the centroid of the portion of the sphere \( x^2 + y^2 + z^2 = 1 \) with \( x \geq 0 \), \( y \geq 0 \), and \( z \geq 0 \). (Assume uniform density.) **5.** Let \( S \) be the surface in the shape of the parabolic bowl \( z = x^2 + y^2 \) with \( z \leq 1 \). Find the flux of \( \mathbf{F} = x \mathbf{i} + y \mathbf{j} \) across \( S \) in the downward direction.
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