If possible, write iterated integrals in spherical coordinates for the following region in the orders dp de dp and de dp dop. Sketch the region of integration. Assume that f is continuous on the region. 2r r/2 SS f(p.4.9)p² sin dp de de 0 1/6 2csc Sketch the region of integration. Choose the correct graph below. O A. O B. r/2 2 O B. Write the integral in spherical coordinates for the given region in the order dp de do. Choose the correct answer below. OA. 2 1/2 2 SSS f(p.4.9)p² sin p dp de do /6 0 2csc 4 r O C. O D.
If possible, write iterated integrals in spherical coordinates for the following region in the orders dp de dp and de dp dop. Sketch the region of integration. Assume that f is continuous on the region. 2r r/2 SS f(p.4.9)p² sin dp de de 0 1/6 2csc Sketch the region of integration. Choose the correct graph below. O A. O B. r/2 2 O B. Write the integral in spherical coordinates for the given region in the order dp de do. Choose the correct answer below. OA. 2 1/2 2 SSS f(p.4.9)p² sin p dp de do /6 0 2csc 4 r O C. O D.
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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![**Integrating in Spherical Coordinates**
In this exercise, we aim to write iterated integrals in spherical coordinates for a specified region. We will explore two different orders: \(d\rho \, d\theta \, d\phi\) and \(d\theta \, d\phi \, d\rho\). Additionally, we will sketch the region of integration, assuming the function \( f \) is continuous on the given region.
**Initial Integral Setup:**
\[
\int_0^{2\pi} \int_{\pi/6}^{\pi/2} \int_0^4 f(\rho, \phi, \theta) \, \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta
\]
**Task 1: Sketch the Region of Integration**
Choose the correct graph depicting the region of integration:
- **Option A:** A partial spherical cap with boundaries extending to \(\rho = 4\), \(\phi = \pi/6\), and \(\phi = \pi/2\).
- **Option B:** A cylindrical shape with height and radius extending to similar bounds.
- **Option C:** A full sphere.
- **Option D:** A toroidal shape with a narrow middle.
**Task 2: Write Integral in Order \(d\theta \, d\phi \, d\rho\)**
Choose the correct expression for the modified integration order:
- **Option A:**
\[
\int_0^4 \int_{\pi/6}^{\pi/2} \int_0^{2\pi} f(\rho, \phi, \theta) \, \rho^2 \sin \phi \, d\theta \, d\phi \, d\rho
\]
- **Option B:**
\[
\int_{\pi/6}^{\pi/2} \int_0^{2\pi} \int_0^4 f(\rho, \phi, \theta) \, \rho^2 \sin \phi \, d\rho \, d\theta \, d\phi
\]
**Graph Explanation:**
The graph for Option A is a three-dimensional partial sphere, bounded by the specified limits in spherical coordinates. The z-axis ranges to 4 with the x and y-axes framed according to traditional Cartesian perspectives. The sliced section](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F511d779c-bf10-4315-98b1-4f0038c55113%2F20332e4a-1bb2-44dd-9091-1efdca8aa7a7%2F5x4qub_processed.png&w=3840&q=75)
Transcribed Image Text:**Integrating in Spherical Coordinates**
In this exercise, we aim to write iterated integrals in spherical coordinates for a specified region. We will explore two different orders: \(d\rho \, d\theta \, d\phi\) and \(d\theta \, d\phi \, d\rho\). Additionally, we will sketch the region of integration, assuming the function \( f \) is continuous on the given region.
**Initial Integral Setup:**
\[
\int_0^{2\pi} \int_{\pi/6}^{\pi/2} \int_0^4 f(\rho, \phi, \theta) \, \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta
\]
**Task 1: Sketch the Region of Integration**
Choose the correct graph depicting the region of integration:
- **Option A:** A partial spherical cap with boundaries extending to \(\rho = 4\), \(\phi = \pi/6\), and \(\phi = \pi/2\).
- **Option B:** A cylindrical shape with height and radius extending to similar bounds.
- **Option C:** A full sphere.
- **Option D:** A toroidal shape with a narrow middle.
**Task 2: Write Integral in Order \(d\theta \, d\phi \, d\rho\)**
Choose the correct expression for the modified integration order:
- **Option A:**
\[
\int_0^4 \int_{\pi/6}^{\pi/2} \int_0^{2\pi} f(\rho, \phi, \theta) \, \rho^2 \sin \phi \, d\theta \, d\phi \, d\rho
\]
- **Option B:**
\[
\int_{\pi/6}^{\pi/2} \int_0^{2\pi} \int_0^4 f(\rho, \phi, \theta) \, \rho^2 \sin \phi \, d\rho \, d\theta \, d\phi
\]
**Graph Explanation:**
The graph for Option A is a three-dimensional partial sphere, bounded by the specified limits in spherical coordinates. The z-axis ranges to 4 with the x and y-axes framed according to traditional Cartesian perspectives. The sliced section
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