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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![### Evaluating a Triple Integral
To evaluate the following integral:
\[
\int_{0}^{1}\int_{0}^{\pi}\int_{\pi/4}^{\pi/2} 24\rho \cos^3\phi \, d\phi \, d\theta \, d\rho
\]
follow these steps:
1. **Integrate with respect to \(\phi\):**
\[
\begin{aligned}
I &= \int_{\pi/4}^{\pi/2} 24\rho \cos^3\phi \, d\phi \\
&= \int_{\pi/4}^{\pi/2} 24\rho (\cos\phi)^3 \, d\phi.
\end{aligned}
\]
2. **Integrate with respect to \(\theta\):**
\[
\begin{aligned}
I &= \int_{0}^{\pi} \left( \int_{\pi/4}^{\pi/2} 24\rho (\cos\phi)^3 \, d\phi \right) d\theta.
\]
3. **Integrate with respect to \(\rho\):**
\[
\begin{aligned}
I &= \int_{0}^{1} \left( \int_{0}^{\pi} \left( \int_{\pi/4}^{\pi/2} 24\rho (\cos\phi)^3 \, d\phi \right) d\theta \right) d\rho.
\]
### Final Answer
Simplify the integrals step by step, ensuring proper application of trigonometric identities and integration techniques.
\[
\int_{0}^{1}\int_{0}^{\pi}\int_{\pi/4}^{\pi/2} 24\rho \cos^3\phi \, d\phi \, d\theta \, d\rho = \boxed{\textrm{(Exact Answer Here)}}
\]
*(Type an exact answer.)*
This section should be completed by substituting the definite integral values and calculating the exact solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1840ebe-b0df-4fe8-9210-d8e3dcfa32cc%2F29b25966-465a-438a-8644-e7c8b7dc4d36%2Fhtrjqe_processed.png&w=3840&q=75)
Transcribed Image Text:### Evaluating a Triple Integral
To evaluate the following integral:
\[
\int_{0}^{1}\int_{0}^{\pi}\int_{\pi/4}^{\pi/2} 24\rho \cos^3\phi \, d\phi \, d\theta \, d\rho
\]
follow these steps:
1. **Integrate with respect to \(\phi\):**
\[
\begin{aligned}
I &= \int_{\pi/4}^{\pi/2} 24\rho \cos^3\phi \, d\phi \\
&= \int_{\pi/4}^{\pi/2} 24\rho (\cos\phi)^3 \, d\phi.
\end{aligned}
\]
2. **Integrate with respect to \(\theta\):**
\[
\begin{aligned}
I &= \int_{0}^{\pi} \left( \int_{\pi/4}^{\pi/2} 24\rho (\cos\phi)^3 \, d\phi \right) d\theta.
\]
3. **Integrate with respect to \(\rho\):**
\[
\begin{aligned}
I &= \int_{0}^{1} \left( \int_{0}^{\pi} \left( \int_{\pi/4}^{\pi/2} 24\rho (\cos\phi)^3 \, d\phi \right) d\theta \right) d\rho.
\]
### Final Answer
Simplify the integrals step by step, ensuring proper application of trigonometric identities and integration techniques.
\[
\int_{0}^{1}\int_{0}^{\pi}\int_{\pi/4}^{\pi/2} 24\rho \cos^3\phi \, d\phi \, d\theta \, d\rho = \boxed{\textrm{(Exact Answer Here)}}
\]
*(Type an exact answer.)*
This section should be completed by substituting the definite integral values and calculating the exact solution.
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