Evaluate the triple integral U= 2x Y "> V = +2 LIK** 2 3 y 2 and w= 72 X f(x, y, z)dxdydz where f(x, y, z)= = x + Triple Integral Region R N 2 به این 3
Evaluate the triple integral U= 2x Y "> V = +2 LIK** 2 3 y 2 and w= 72 X f(x, y, z)dxdydz where f(x, y, z)= = x + Triple Integral Region R N 2 به این 3
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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![**Instructions: Evaluating an Integral Using the Jacobian**
Remember that:
\[
\iiint\limits_R F(x, y, z) dV = \iiint\limits_G H(u, v, w) |J(u, v, w)| dudvdw
\]
- **\( u \) lower limit:**
- **\( u \) upper limit:**
- **\( v \) lower limit:**
- **\( v \) upper limit:**
- **\( w \) lower limit:**
- **\( w \) upper limit:**
**\( H(u, v, w) = \)**
**\(|J(u, v, w)| = \)**
\[
\iiint\limits_G H(u, v, w) |J(u, v, w)| dudvdw =
\]
*Hint:* The focus of this problem is on evaluating the integral and using the Jacobian.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd20dfe5a-a4c1-4793-9f05-a80ad59a67d4%2Fff0618d5-ab5a-4dce-a8a9-fd5fa2b69e3e%2Fbp2izme_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Instructions: Evaluating an Integral Using the Jacobian**
Remember that:
\[
\iiint\limits_R F(x, y, z) dV = \iiint\limits_G H(u, v, w) |J(u, v, w)| dudvdw
\]
- **\( u \) lower limit:**
- **\( u \) upper limit:**
- **\( v \) lower limit:**
- **\( v \) upper limit:**
- **\( w \) lower limit:**
- **\( w \) upper limit:**
**\( H(u, v, w) = \)**
**\(|J(u, v, w)| = \)**
\[
\iiint\limits_G H(u, v, w) |J(u, v, w)| dudvdw =
\]
*Hint:* The focus of this problem is on evaluating the integral and using the Jacobian.
![### Triple Integral Evaluation over a Region
**Problem Statement:**
Evaluate the triple integral:
\[
\int_{0}^{1} \int_{0}^{1} \int_{\frac{y}{2}}^{\frac{y}{2} + 2} f(x, y, z) \, dx \, dy \, dz
\]
where
\[
f(x, y, z) = x + \frac{z}{3}
\]
with the transformations:
\[
u = \frac{2x - y}{2}, \quad v = \frac{y}{2}, \quad w = \frac{z}{3}
\]
**Graph Explanation:**
The image includes a 3D diagram representing the region \( R \) over which the triple integral is evaluated. This region appears as a rectangular prism within the 3D coordinate system.
- **Axes:**
- The \( x \)-axis ranges from 1 to 3.
- The \( y \)-axis ranges from 0 to 4.
- The \( z \)-axis ranges from 0 to 3.
- **Region Description:**
- The prism is aligned along the \( z \)-axis, demonstrating the bounds of integration with respect to \( x, y, \) and \( z \).
- The sides of the prism are color-coded for clarity, usually differentiated to represent the boundary planes of the region of integration.
**Conclusion:**
This visual aids in understanding how each variable \( x, y, \) and \( z \) changes across their specified intervals in space, crucial for setting up and solving a triple integral in multivariable calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd20dfe5a-a4c1-4793-9f05-a80ad59a67d4%2Fff0618d5-ab5a-4dce-a8a9-fd5fa2b69e3e%2Fyav2rb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Triple Integral Evaluation over a Region
**Problem Statement:**
Evaluate the triple integral:
\[
\int_{0}^{1} \int_{0}^{1} \int_{\frac{y}{2}}^{\frac{y}{2} + 2} f(x, y, z) \, dx \, dy \, dz
\]
where
\[
f(x, y, z) = x + \frac{z}{3}
\]
with the transformations:
\[
u = \frac{2x - y}{2}, \quad v = \frac{y}{2}, \quad w = \frac{z}{3}
\]
**Graph Explanation:**
The image includes a 3D diagram representing the region \( R \) over which the triple integral is evaluated. This region appears as a rectangular prism within the 3D coordinate system.
- **Axes:**
- The \( x \)-axis ranges from 1 to 3.
- The \( y \)-axis ranges from 0 to 4.
- The \( z \)-axis ranges from 0 to 3.
- **Region Description:**
- The prism is aligned along the \( z \)-axis, demonstrating the bounds of integration with respect to \( x, y, \) and \( z \).
- The sides of the prism are color-coded for clarity, usually differentiated to represent the boundary planes of the region of integration.
**Conclusion:**
This visual aids in understanding how each variable \( x, y, \) and \( z \) changes across their specified intervals in space, crucial for setting up and solving a triple integral in multivariable calculus.
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