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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Question
![**Problem Statement**
Use integration in spherical coordinates to obtain the general formula for the volume of a ball with radius \( R \).
**Solution Outline**
To find the volume of a ball using spherical coordinates, consider the following steps:
1. **Define Spherical Coordinates**: In spherical coordinates, a point in space is defined by three parameters: the radial distance \( r \), the polar angle \( \theta \) (usually measured from the positive z-axis), and the azimuthal angle \( \phi \) (measured from the positive x-axis).
2. **Volume Element**: The volume element in spherical coordinates is given by \( dV = r^2 \sin \theta \, dr \, d\theta \, d\phi \).
3. **Integration Limits**:
- \( r \) ranges from 0 to \( R \).
- \( \theta \) ranges from 0 to \( \pi \).
- \( \phi \) ranges from 0 to \( 2\pi \).
4. **Setup the Integral**: The volume \( V \) of the ball can be calculated with the integral:
\[
V = \int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{R} r^2 \sin \theta \, dr \, d\theta \, d\phi
\]
5. **Solve the Integral**: Evaluate the triple integral step-by-step to find the volume.
**Outcome**
The result of this integration will yield the familiar formula for the volume of a sphere:
\[
V = \frac{4}{3}\pi R^3
\]
This exercise demonstrates the utility of spherical coordinates in simplifying the computation of volumes for symmetrical 3-dimensional shapes like spheres.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa923b6f-81dd-482c-8885-6de6bc295751%2F57cc7af4-fcef-4132-a656-06320f488b59%2F1w56bzs_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Use integration in spherical coordinates to obtain the general formula for the volume of a ball with radius \( R \).
**Solution Outline**
To find the volume of a ball using spherical coordinates, consider the following steps:
1. **Define Spherical Coordinates**: In spherical coordinates, a point in space is defined by three parameters: the radial distance \( r \), the polar angle \( \theta \) (usually measured from the positive z-axis), and the azimuthal angle \( \phi \) (measured from the positive x-axis).
2. **Volume Element**: The volume element in spherical coordinates is given by \( dV = r^2 \sin \theta \, dr \, d\theta \, d\phi \).
3. **Integration Limits**:
- \( r \) ranges from 0 to \( R \).
- \( \theta \) ranges from 0 to \( \pi \).
- \( \phi \) ranges from 0 to \( 2\pi \).
4. **Setup the Integral**: The volume \( V \) of the ball can be calculated with the integral:
\[
V = \int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{R} r^2 \sin \theta \, dr \, d\theta \, d\phi
\]
5. **Solve the Integral**: Evaluate the triple integral step-by-step to find the volume.
**Outcome**
The result of this integration will yield the familiar formula for the volume of a sphere:
\[
V = \frac{4}{3}\pi R^3
\]
This exercise demonstrates the utility of spherical coordinates in simplifying the computation of volumes for symmetrical 3-dimensional shapes like spheres.
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