2x dV, where E is bounded by the paraboloid x = 2y2 + 2z2 and the plane x = 2. %3D

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Question

Evaluate the triple integral.

The integral to evaluate is given by:

\[
\iiint\limits_{E} 2x \, dV
\]

where \( E \) is the region bounded by the paraboloid \( x = 2y^2 + 2z^2 \) and the plane \( x = 2 \).

### Explanation:

The expression represents a triple integral over a region \( E \) in three-dimensional space. The function \( 2x \) is integrated with respect to the volume element \( dV \).

- **Region \( E \):** 
  - **Paraboloid:** Described by the equation \( x = 2y^2 + 2z^2 \), a surface where \( x \) depends on the squares of \( y \) and \( z \).
  - **Plane:** The plane \( x = 2 \) acts as an upper bound for the region of integration.

The objective is to calculate the volume integral of the function \( 2x \) within these boundaries.
Transcribed Image Text:The integral to evaluate is given by: \[ \iiint\limits_{E} 2x \, dV \] where \( E \) is the region bounded by the paraboloid \( x = 2y^2 + 2z^2 \) and the plane \( x = 2 \). ### Explanation: The expression represents a triple integral over a region \( E \) in three-dimensional space. The function \( 2x \) is integrated with respect to the volume element \( dV \). - **Region \( E \):** - **Paraboloid:** Described by the equation \( x = 2y^2 + 2z^2 \), a surface where \( x \) depends on the squares of \( y \) and \( z \). - **Plane:** The plane \( x = 2 \) acts as an upper bound for the region of integration. The objective is to calculate the volume integral of the function \( 2x \) within these boundaries.
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