Evaluate the triple integral U= 21-y 2 " V= Y 2 and w== 22 3 f(x, y, z)dxdydz where f(x, y, z) = 1 + N 100
Evaluate the triple integral U= 21-y 2 " V= Y 2 and w== 22 3 f(x, y, z)dxdydz where f(x, y, z) = 1 + N 100
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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![### Evaluating a Triple Integral
Consider the following triple integral which we are required to evaluate:
\[ \iiint_{0}^{\frac{7y}{2} + 5} \int_{0}^{3} \int_{0}^{1} f(x, y, z) \, dx \, dy \, dz \]
where \( f(x, y, z) = x + \frac{z}{3} \).
Given the transformations:
\[
u = \frac{2x - y}{2}, \quad v = \frac{y}{2}, \quad w = \frac{z}{3}.
\]
#### Visualization of Region \( R \)
The triple integral is taken over a region \( R \) which can be visualized as a three-dimensional rectangular prism as shown in the diagram:
In this 3D plot:
- \( x \), \( y \), and \( z \) axes are labeled.
- The region \( R \) is bounded by:
- \( x \) ranging from 0 to 3
- \( y \) from 0 to 1
- \( z \) from 0 to \( \frac{7y}{2} + 5 \)
### Transformations from \((x, y, z)\) to \((u, v, w)\)
Remember that:
\[
\iiint_{R} F(x, y, z) \, dV = \iiint_{G} H(u, v, w) |J(u, v, w)| \, du \, dv \, dw
\]
To correctly transform the limits of integration:
#### Lower and Upper Limits for \( u \), \( v \), and \( w \)
- **\( u \) lower limit** =
- **\( u \) upper limit** =
- **\( v \) lower limit** =
- **\( v \) upper limit** =
- **\( w \) lower limit** =
- **\( w \) upper limit** =
#### Function and Jacobian
- **\( H(u, v, w) \) =**
- **\( |J(u, v, w)| \) =**
Finally, transform and evaluate the triple integral:
\[
\iiint_{G} H(u, v](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4100c518-1a8f-4898-88a2-d0f1a7669694%2F53b8a3aa-2dd3-4808-b444-1e07c7dcb1bc%2Fv6fukr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Evaluating a Triple Integral
Consider the following triple integral which we are required to evaluate:
\[ \iiint_{0}^{\frac{7y}{2} + 5} \int_{0}^{3} \int_{0}^{1} f(x, y, z) \, dx \, dy \, dz \]
where \( f(x, y, z) = x + \frac{z}{3} \).
Given the transformations:
\[
u = \frac{2x - y}{2}, \quad v = \frac{y}{2}, \quad w = \frac{z}{3}.
\]
#### Visualization of Region \( R \)
The triple integral is taken over a region \( R \) which can be visualized as a three-dimensional rectangular prism as shown in the diagram:
In this 3D plot:
- \( x \), \( y \), and \( z \) axes are labeled.
- The region \( R \) is bounded by:
- \( x \) ranging from 0 to 3
- \( y \) from 0 to 1
- \( z \) from 0 to \( \frac{7y}{2} + 5 \)
### Transformations from \((x, y, z)\) to \((u, v, w)\)
Remember that:
\[
\iiint_{R} F(x, y, z) \, dV = \iiint_{G} H(u, v, w) |J(u, v, w)| \, du \, dv \, dw
\]
To correctly transform the limits of integration:
#### Lower and Upper Limits for \( u \), \( v \), and \( w \)
- **\( u \) lower limit** =
- **\( u \) upper limit** =
- **\( v \) lower limit** =
- **\( v \) upper limit** =
- **\( w \) lower limit** =
- **\( w \) upper limit** =
#### Function and Jacobian
- **\( H(u, v, w) \) =**
- **\( |J(u, v, w)| \) =**
Finally, transform and evaluate the triple integral:
\[
\iiint_{G} H(u, v
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