Use spherical coordinates to evaluate the triple integral e-(x² + y² +₂²) x² + y² + z² III. E dV, where E is the region bounded by the spheres x² + y² + z² = 4 and x² + y² + z² = 16.
Use spherical coordinates to evaluate the triple integral e-(x² + y² +₂²) x² + y² + z² III. E dV, where E is the region bounded by the spheres x² + y² + z² = 4 and x² + y² + z² = 16.
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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![**Triple Integral Evaluation Using Spherical Coordinates**
Evaluate the triple integral using spherical coordinates:
\[
\iiint_{E} \frac{e^{-(x^2 + y^2 + z^2)}}{\sqrt{x^2 + y^2 + z^2}} \, dV,
\]
where \( E \) is the region bounded by the spheres \( x^2 + y^2 + z^2 = 4 \) and \( x^2 + y^2 + z^2 = 16 \).
**Answer:**
\[
2\pi \left[ e^{-16} - e^{-4} \right]
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Transcribed Image Text:**Triple Integral Evaluation Using Spherical Coordinates**
Evaluate the triple integral using spherical coordinates:
\[
\iiint_{E} \frac{e^{-(x^2 + y^2 + z^2)}}{\sqrt{x^2 + y^2 + z^2}} \, dV,
\]
where \( E \) is the region bounded by the spheres \( x^2 + y^2 + z^2 = 4 \) and \( x^2 + y^2 + z^2 = 16 \).
**Answer:**
\[
2\pi \left[ e^{-16} - e^{-4} \right]
\]
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