Consider the following two solids: i) the ball of radius 3 centered at the origin and ii) the cone defined by the condition ø < ™/6 where ø is the spherical coordinate with the same symbol. Their intersection looks like an ice-cream cone. Use integration with spherical coordinates in order to compute the volume of this intersection.
Consider the following two solids: i) the ball of radius 3 centered at the origin and ii) the cone defined by the condition ø < ™/6 where ø is the spherical coordinate with the same symbol. Their intersection looks like an ice-cream cone. Use integration with spherical coordinates in order to compute the volume of this intersection.
Calculus: Early Transcendentals
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ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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![**Problem Statement: Volume Intersection Using Spherical Coordinates**
**Concept Overview:**
Consider the following two solids:
1. A ball of radius 3 centered at the origin.
2. A cone defined by the condition \( \phi \leq \pi/6 \), where \( \phi \) is the spherical coordinate of the same symbol.
The intersection of these two solids resembles an ice-cream cone.
**Objective:**
Use integration with spherical coordinates to compute the volume of this intersection.
**Explanation and Steps:**
To solve this problem, we need to understand the spherical coordinate system, where:
- \( \rho \) (rho) is the radial distance from the origin.
- \( \phi \) (phi) is the angle from the positive z-axis.
- \( \theta \) (theta) is the azimuthal angle in the xy-plane from the positive x-axis.
**Integration Setup:**
For the specified solids:
- The radial distance \(\rho\) will vary from 0 to 3.
- The angle \(\phi\) will vary from 0 to \(\pi/6\).
- The angle \(\theta\) will vary from 0 to \(2\pi\) to cover the entire circular symmetry of the cone.
The volume element in spherical coordinates is given by:
\[ dV = \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta \]
**Volume Calculation:**
The integral for the volume \( V \) can be set up as:
\[ V = \int_{\theta=0}^{2\pi} \int_{\phi=0}^{\pi/6} \int_{\rho=0}^{3} \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta \]
By evaluating the above integral, one can compute the volume of the intersection that resembles an ice-cream cone.
**Conclusion:**
This method uses the principles of spherical coordinates to accurately determine the volume of the complex intersection between a sphere and a cone, demonstrating the power of integration in solving geometric problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa923b6f-81dd-482c-8885-6de6bc295751%2Fd70fda7c-d282-4d89-bc1a-c060d089f8a8%2Fgbr51mp_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Volume Intersection Using Spherical Coordinates**
**Concept Overview:**
Consider the following two solids:
1. A ball of radius 3 centered at the origin.
2. A cone defined by the condition \( \phi \leq \pi/6 \), where \( \phi \) is the spherical coordinate of the same symbol.
The intersection of these two solids resembles an ice-cream cone.
**Objective:**
Use integration with spherical coordinates to compute the volume of this intersection.
**Explanation and Steps:**
To solve this problem, we need to understand the spherical coordinate system, where:
- \( \rho \) (rho) is the radial distance from the origin.
- \( \phi \) (phi) is the angle from the positive z-axis.
- \( \theta \) (theta) is the azimuthal angle in the xy-plane from the positive x-axis.
**Integration Setup:**
For the specified solids:
- The radial distance \(\rho\) will vary from 0 to 3.
- The angle \(\phi\) will vary from 0 to \(\pi/6\).
- The angle \(\theta\) will vary from 0 to \(2\pi\) to cover the entire circular symmetry of the cone.
The volume element in spherical coordinates is given by:
\[ dV = \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta \]
**Volume Calculation:**
The integral for the volume \( V \) can be set up as:
\[ V = \int_{\theta=0}^{2\pi} \int_{\phi=0}^{\pi/6} \int_{\rho=0}^{3} \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta \]
By evaluating the above integral, one can compute the volume of the intersection that resembles an ice-cream cone.
**Conclusion:**
This method uses the principles of spherical coordinates to accurately determine the volume of the complex intersection between a sphere and a cone, demonstrating the power of integration in solving geometric problems.
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