Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**In Spherical Coordinates:**
The image above is a diagram illustrating spherical coordinates. It shows a point \( M(x, y, z) \) in 3D space, represented in terms of spherical coordinates \((\rho, \phi, \theta)\). The coordinates are defined as follows:
- \( \rho \) is the radial distance from the origin \( O \) to the point \( M \).
- \( \phi \) (phi) is the angle between the positive \( z \)-axis and the line \( OM \).
- \( \theta \) (theta) is the angle between the positive \( x \)-axis and the projection of line \( OM \) on the \( xy \)-plane.
The diagram includes:
- An arrow from the origin to the point \( M(x, y, z) \) labeled with \( \rho \).
- Angle \( \phi \) is shown between the \( z \)-axis and \( \rho \).
- Angle \( \theta \) is shown between the line's projection in the \( xy \)-plane and the \( x \)-axis.
**Using the formula for the change of variables in triple integrals:**
\[
\iiint_U f(x, y, z) \, dx \, dy \, dz = \iiint_{U'} f(\varphi, \psi, \chi) \, |J(u, v, w)| \, du \, dv \, dw,
\]
where \( |J(u, v, w)| \) means the absolute value of the Jacobian.
**Task:**
Derive the triple integral in Spherical coordinates using the “change of variables” formula (you don’t have to INTEGRATE!).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda21d2aa-bb54-4a64-8baf-73046aeb648e%2F097f4a3d-5bd2-426a-9d91-4944e031fd0d%2Fvgbinx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**In Spherical Coordinates:**
The image above is a diagram illustrating spherical coordinates. It shows a point \( M(x, y, z) \) in 3D space, represented in terms of spherical coordinates \((\rho, \phi, \theta)\). The coordinates are defined as follows:
- \( \rho \) is the radial distance from the origin \( O \) to the point \( M \).
- \( \phi \) (phi) is the angle between the positive \( z \)-axis and the line \( OM \).
- \( \theta \) (theta) is the angle between the positive \( x \)-axis and the projection of line \( OM \) on the \( xy \)-plane.
The diagram includes:
- An arrow from the origin to the point \( M(x, y, z) \) labeled with \( \rho \).
- Angle \( \phi \) is shown between the \( z \)-axis and \( \rho \).
- Angle \( \theta \) is shown between the line's projection in the \( xy \)-plane and the \( x \)-axis.
**Using the formula for the change of variables in triple integrals:**
\[
\iiint_U f(x, y, z) \, dx \, dy \, dz = \iiint_{U'} f(\varphi, \psi, \chi) \, |J(u, v, w)| \, du \, dv \, dw,
\]
where \( |J(u, v, w)| \) means the absolute value of the Jacobian.
**Task:**
Derive the triple integral in Spherical coordinates using the “change of variables” formula (you don’t have to INTEGRATE!).
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