Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Problem Description:**
Use spherical coordinates.
**Evaluate the triple integral:**
\[
\iiint_{E} (x^2 + y^2) \, dV
\]
where \( E \) lies between the spheres \( x^2 + y^2 + z^2 = 1 \) and \( x^2 + y^2 + z^2 = 9 \).
**Explanation:**
This problem requires evaluating a triple integral over the region \( E \) that is confined between two concentric spheres. The inner sphere has a radius of 1 (given by \( x^2 + y^2 + z^2 = 1 \)) and the outer sphere has a radius of 3 (given by \( x^2 + y^2 + z^2 = 9 \)).
To solve this integral, spherical coordinates should be used:
- The spherical coordinate transformations are provided by:
- \( x = \rho \sin\phi \cos\theta \)
- \( y = \rho \sin\phi \sin\theta \)
- \( z = \rho \cos\phi \)
- The volume element \( dV \) in spherical coordinates becomes:
- \( dV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta \)
- The limits for \( \rho \) are from 1 to 3, \( \phi \) from 0 to \( \pi \), and \( \theta \) from 0 to \( 2\pi \).
By substituting \( x \) and \( y \) as described above into the integrand \( x^2 + y^2 \), and using the volume element in spherical coordinates, you can set up the integral to evaluate over the specified region.](https://content.bartleby.com/qna-images/question/9a80861c-f7b1-4a5c-9dd8-fa536dc93a77/1f16f7da-52ff-4d8f-ad2d-e10863da1402/xeg7lxl_processed.jpeg)
Transcribed Image Text:**Problem Description:**
Use spherical coordinates.
**Evaluate the triple integral:**
\[
\iiint_{E} (x^2 + y^2) \, dV
\]
where \( E \) lies between the spheres \( x^2 + y^2 + z^2 = 1 \) and \( x^2 + y^2 + z^2 = 9 \).
**Explanation:**
This problem requires evaluating a triple integral over the region \( E \) that is confined between two concentric spheres. The inner sphere has a radius of 1 (given by \( x^2 + y^2 + z^2 = 1 \)) and the outer sphere has a radius of 3 (given by \( x^2 + y^2 + z^2 = 9 \)).
To solve this integral, spherical coordinates should be used:
- The spherical coordinate transformations are provided by:
- \( x = \rho \sin\phi \cos\theta \)
- \( y = \rho \sin\phi \sin\theta \)
- \( z = \rho \cos\phi \)
- The volume element \( dV \) in spherical coordinates becomes:
- \( dV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta \)
- The limits for \( \rho \) are from 1 to 3, \( \phi \) from 0 to \( \pi \), and \( \theta \) from 0 to \( 2\pi \).
By substituting \( x \) and \( y \) as described above into the integrand \( x^2 + y^2 \), and using the volume element in spherical coordinates, you can set up the integral to evaluate over the specified region.
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Evaluate the triple integrals conversion yo spherical co-ordinates
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