5. f(x, y, z) = x z=1-x² X ZA y+z=2 E y
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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Complete problem 5
![### Evaluating Triple Integrals
To evaluate the triple integral \(\iiint_E f(x, y, z) \, dV\) as an iterated integral for the given function \(f\) and the solid region \(E\):
#### Functions and Regions
1. **Function 5: \(f(x, y, z) = x\)**
- Region \(E\) is defined by:
- **Bounds**:
- \(z = 1 - x^2\)
- \(y + z = 2\)
- **Diagram**: Displays a region in a 3D space enclosed between the given surfaces.
2. **Function 6: \(f(x, y, z) = xy\)**
- Region \(E\) is defined by:
- **Bounds**:
- \(z = 4 - y^2\)
- \(y = x\)
- **Diagram**: Illustrates another solid region where the boundaries form a curved shape in a 3-dimensional coordinate system.
#### Coordinate Transformation
Use the transformation \(u = x + y\), \(v = x - y\) to set up the iterated integral
\[
\iint_R \frac{x-y}{x+y} \, dA
\]
- **Region \(R\)** has vertices at:
- \((0, 2)\)
- \((1, 1)\)
- \((2, 2)\)
- \((1, 3)\)
These transformations and expressions are used to simplify the integrals over the region \(R\) using the new coordinates \((u, v)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F76c78a43-228d-485d-a6cf-d30956c84b3b%2F1ad5c3ed-cb29-441a-bd85-4eb487d8c148%2Fv55bi4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Evaluating Triple Integrals
To evaluate the triple integral \(\iiint_E f(x, y, z) \, dV\) as an iterated integral for the given function \(f\) and the solid region \(E\):
#### Functions and Regions
1. **Function 5: \(f(x, y, z) = x\)**
- Region \(E\) is defined by:
- **Bounds**:
- \(z = 1 - x^2\)
- \(y + z = 2\)
- **Diagram**: Displays a region in a 3D space enclosed between the given surfaces.
2. **Function 6: \(f(x, y, z) = xy\)**
- Region \(E\) is defined by:
- **Bounds**:
- \(z = 4 - y^2\)
- \(y = x\)
- **Diagram**: Illustrates another solid region where the boundaries form a curved shape in a 3-dimensional coordinate system.
#### Coordinate Transformation
Use the transformation \(u = x + y\), \(v = x - y\) to set up the iterated integral
\[
\iint_R \frac{x-y}{x+y} \, dA
\]
- **Region \(R\)** has vertices at:
- \((0, 2)\)
- \((1, 1)\)
- \((2, 2)\)
- \((1, 3)\)
These transformations and expressions are used to simplify the integrals over the region \(R\) using the new coordinates \((u, v)\).
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