Evaluate the triple integral. 3x dV, where E = {(x, y, z) I 0 s y s 2, 0sxs V4- y², 0 s z s 2y}
Evaluate the triple integral. 3x dV, where E = {(x, y, z) I 0 s y s 2, 0sxs V4- y², 0 s z s 2y}
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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![Evaluate the triple integral.
\[
\iiint\limits_{E} 3x \, dV
\]
where \( E = \{ (x, y, z) \mid 0 \leq y \leq 2, 0 \leq x \leq \sqrt{4 - y^2}, 0 \leq z \leq 2y \} \).
**Explanation:**
- The integral is a triple integral over the region \( E \).
- The region \( E \) is defined in a three-dimensional space with variables \( x \), \( y \), and \( z \).
- The limits for \( y \) are from 0 to 2.
- The limits for \( x \) are from 0 to \( \sqrt{4 - y^2} \), which describes a semicircle in the \( xy \)-plane.
- The limits for \( z \) are from 0 to \( 2y \), indicating that the value of \( z \) depends linearly on \( y \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d3987b5-d473-4459-9b6d-f1c57ff7aeed%2F09f71500-1b2a-4c25-b2fd-416c99097a85%2Fsql8p9d_processed.png&w=3840&q=75)
Transcribed Image Text:Evaluate the triple integral.
\[
\iiint\limits_{E} 3x \, dV
\]
where \( E = \{ (x, y, z) \mid 0 \leq y \leq 2, 0 \leq x \leq \sqrt{4 - y^2}, 0 \leq z \leq 2y \} \).
**Explanation:**
- The integral is a triple integral over the region \( E \).
- The region \( E \) is defined in a three-dimensional space with variables \( x \), \( y \), and \( z \).
- The limits for \( y \) are from 0 to 2.
- The limits for \( x \) are from 0 to \( \sqrt{4 - y^2} \), which describes a semicircle in the \( xy \)-plane.
- The limits for \( z \) are from 0 to \( 2y \), indicating that the value of \( z \) depends linearly on \( y \).
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