The image provided contains a problem related to multiple integrals in a spherical coordinate system.### Diagram Explanation:The diagram shows a cone with its vertex at the origin, oriented along the positive z-axis, with a circular base on the xy-plane. The cone is capped by the top half of a sphere with radius 2, centered at the origin.### Problem Statement:Let $ E $ represent the region bounded below the cone $ z = \sqrt{x^2 + y^2} $, inside the sphere $ x^2 + y^2 + z^2 = 4 $, and above the plane $ z = 0 $ (see diagram above). Choose the iterated integral that is equivalent to $\int_E 4z \, dV$.### Multiple Choice Options:1. \[ \int_0^2 \int_0^{\pi/4} \int_0^{2\pi} 4\rho^3 \sin \phi \cos \phi \, d\theta \, d\phi \, d\rho \]2. \[ \int_0^2 \int_{\pi/4}^{\pi/2} \int_0^{2\pi} 4\rho^3 \sin \phi \cos \phi \, d\theta \, d\phi \, d\rho \]3. \[ \int_0^{\sqrt{2}} \int_{\pi/4}^{\pi/2} \int_0^{2\pi} 4\rho^3 \sin \phi \cos \phi \, d\theta \, d\phi \, d\rho \]4. \[ \int_0^{\sqrt{2}} \int_0^{\pi/4} \int_0^{2\pi} 4\rho^3 \sin \phi \cos \phi \, d\theta \, d\phi \, d\rho \]

Answered: Image /qna-images/answer/51d02341-7bc1-437c-9885-3682edfbadfd.jpg

(x² + y?, inside 0 (see diagram Let E represent the region bounded below the cone z = the sphere x2 +y? + z? = 4, and above the plane z = above). Choose the iterated integral that is equivalent to 4z dV.

(x² + y?, inside 0 (see diagram Let E represent the region bounded below the cone z = the sphere x2 +y? + z? = 4, and above the plane z = above). Choose the iterated integral that is equivalent to 4z dV.

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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Question

The image provided contains a problem related to multiple integrals in a spherical coordinate system.

### Diagram Explanation:
The diagram shows a cone with its vertex at the origin, oriented along the positive z-axis, with a circular base on the xy-plane. The cone is capped by the top half of a sphere with radius 2, centered at the origin.

### Problem Statement:
Let $ E $ represent the region bounded below the cone $ z = \sqrt{x^2 + y^2} $, inside the sphere $ x^2 + y^2 + z^2 = 4 $, and above the plane $ z = 0 $ (see diagram above). Choose the iterated integral that is equivalent to $\int_E 4z \, dV$.

### Multiple Choice Options:
1. \[
\int_0^2 \int_0^{\pi/4} \int_0^{2\pi} 4\rho^3 \sin \phi \cos \phi \, d\theta \, d\phi \, d\rho
\]

2. \[
\int_0^2 \int_{\pi/4}^{\pi/2} \int_0^{2\pi} 4\rho^3 \sin \phi \cos \phi \, d\theta \, d\phi \, d\rho
\]

3. \[
\int_0^{\sqrt{2}} \int_{\pi/4}^{\pi/2} \int_0^{2\pi} 4\rho^3 \sin \phi \cos \phi \, d\theta \, d\phi \, d\rho
\]

4. \[
\int_0^{\sqrt{2}} \int_0^{\pi/4} \int_0^{2\pi} 4\rho^3 \sin \phi \cos \phi \, d\theta \, d\phi \, d\rho
\]

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