Set up the triple integral of an arbitrary continuous function f(x, y, z) in spherical coordinates over the solid shown. (Assume a = 1 and b = 7.) SIS f(x, y, z) dv= Jπ/2 dp de dep 10
Set up the triple integral of an arbitrary continuous function f(x, y, z) in spherical coordinates over the solid shown. (Assume a = 1 and b = 7.) SIS f(x, y, z) dv= Jπ/2 dp de dep 10
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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![### Understanding Triple Integrals in Spherical Coordinates
This educational module is designed to help you set up a triple integral of an arbitrary continuous function \( f(x, y, z) \) in spherical coordinates over a given solid region.
#### Problem Setup
Given the integral:
\[
\iiint_{E} f(x, y, z) \, dV
\]
We aim to express this integral in spherical coordinates. Assume the following parameters:
- \( a = 1 \)
- \( b = 7 \)
#### Integral Expression
The triple integral in spherical coordinates can be set up as:
\[
\int_{?}^{?} \left( \int_{0}^{?} \left( \int_{1}^{?} f \left( ?, ?, ? \right) \, d\rho \right) d\theta \right) d\phi
\]
Here, \( \rho \), \( \theta \), and \( \phi \) are the spherical coordinates.
#### Diagram Description
- **3D Representation:** The diagram shows a solid bounded by two hemispheres.
- These hemispheres have radii of \( a \) and \( b \), where \( a = 1 \) and \( b = 7 \).
- The solid lies above the \( xy \)-plane.
- **Coordinates:**
- Origin at the intersection of the three axes.
- Axes labeled \( x \), \( y \), and \( z \).
- **Region in 3D:** The area in blue represents the volume over which we will be integrating.
#### Step-by-Step Integration Limits
1. **Innermost Integral (with respect to \( \rho \)):** This defines the radial distance from the origin.
- Limits for \( \rho \): From \( a \) to \( b \).
2. **Middle Integral (with respect to \( \theta \)):** This span the azimuthal angle in the \( xy \)-plane.
- Limits for \( \theta \): From 0 to \( \pi/2 \).
3. **Outermost Integral (with respect to \( \phi \)):** This defines the polar angle.
- Limits for \( \phi \): From 1 to the unknown upper limit (typically \( \pi \) for a full hemisphere).
#### Writing the Function \( f \)
In spherical coordinates:
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed20e167-a048-49f1-8a38-9bdf52d30521%2F10610291-5930-41a0-92f8-39fd7dc4c9e4%2Fvmd5hag_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Triple Integrals in Spherical Coordinates
This educational module is designed to help you set up a triple integral of an arbitrary continuous function \( f(x, y, z) \) in spherical coordinates over a given solid region.
#### Problem Setup
Given the integral:
\[
\iiint_{E} f(x, y, z) \, dV
\]
We aim to express this integral in spherical coordinates. Assume the following parameters:
- \( a = 1 \)
- \( b = 7 \)
#### Integral Expression
The triple integral in spherical coordinates can be set up as:
\[
\int_{?}^{?} \left( \int_{0}^{?} \left( \int_{1}^{?} f \left( ?, ?, ? \right) \, d\rho \right) d\theta \right) d\phi
\]
Here, \( \rho \), \( \theta \), and \( \phi \) are the spherical coordinates.
#### Diagram Description
- **3D Representation:** The diagram shows a solid bounded by two hemispheres.
- These hemispheres have radii of \( a \) and \( b \), where \( a = 1 \) and \( b = 7 \).
- The solid lies above the \( xy \)-plane.
- **Coordinates:**
- Origin at the intersection of the three axes.
- Axes labeled \( x \), \( y \), and \( z \).
- **Region in 3D:** The area in blue represents the volume over which we will be integrating.
#### Step-by-Step Integration Limits
1. **Innermost Integral (with respect to \( \rho \)):** This defines the radial distance from the origin.
- Limits for \( \rho \): From \( a \) to \( b \).
2. **Middle Integral (with respect to \( \theta \)):** This span the azimuthal angle in the \( xy \)-plane.
- Limits for \( \theta \): From 0 to \( \pi/2 \).
3. **Outermost Integral (with respect to \( \phi \)):** This defines the polar angle.
- Limits for \( \phi \): From 1 to the unknown upper limit (typically \( \pi \) for a full hemisphere).
#### Writing the Function \( f \)
In spherical coordinates:
\[
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