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
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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 \( y \).
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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