SI. f dV here D is the solid region in the first octant between the elliptic cylinder 4x2 + 22 = 16 an ane y = 3. (0,0, 4) 4.x2 + z2 = 16 (0,3, 4) (2,0,0)
SI. f dV here D is the solid region in the first octant between the elliptic cylinder 4x2 + 22 = 16 an ane y = 3. (0,0, 4) 4.x2 + z2 = 16 (0,3, 4) (2,0,0)
SI. f dV here D is the solid region in the first octant between the elliptic cylinder 4x2 + 22 = 16 an ane y = 3. (0,0, 4) 4.x2 + z2 = 16 (0,3, 4) (2,0,0)
Transcribed Image Text:### Problem Statement
Let \( f(x, y, z) = xy \). Evaluate
\[
\iiint_D f \, dV
\]
where \(D\) is the solid region in the first octant between the elliptic cylinder \( 4x^2 + z^2 = 16 \) and the plane \( y = 3 \).
### Diagram Explanation
The provided diagram visually represents the solid region \( D \) in the first octant.
#### Key Features:
- **Elliptic Cylinder:** Represented by the equation \( 4x^2 + z^2 = 16 \). This surface is depicted as the curved, blue-shaded region in the three-dimensional plot.
- **Plane \( y = 3 \):** This plane is depicted as the gray-shaded region. It intersects the elliptic cylinder and confines the region of integration.
#### Axes:
- **x-axis:** Extends from \( (2,0,0) \) horizontally.
- **y-axis:** Extends from \( (2,3,0) \) vertically.
- **z-axis:** Extends from \( (0,0,4) \) upwards.
#### Intersections:
- **Intersection of Cylinder and Plane (at boundaries):**
- At \( y=0 \):
- Points: \( (2, 0, 0) \) and \( (0, 0, 4) \)
- At \( y=3 \):
- Points: \( (2, 3, 0) \) and \( (0, 3, 4) \)
- The intersection points of the elliptic cylinder with the plane \( y = 3 \) form a rectangular boundary for the region in consideration.
The diagram is useful for visualizing the limits and boundaries of the region \( D \) we are integrating over. It highlights the interaction of a three-dimensional solid constrained by both a curved surface (the elliptic cylinder) and a flat surface (the plane \(y=3\)).
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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![### Problem Statement
Let \( f(x, y, z) = xy \). Evaluate
\[
\iiint_D f \, dV
\]
where \(D\) is the solid region in the first octant between the elliptic cylinder \( 4x^2 + z^2 = 16 \) and the plane \( y = 3 \).
### Diagram Explanation
The provided diagram visually represents the solid region \( D \) in the first octant.
#### Key Features:
- **Elliptic Cylinder:** Represented by the equation \( 4x^2 + z^2 = 16 \). This surface is depicted as the curved, blue-shaded region in the three-dimensional plot.
- **Plane \( y = 3 \):** This plane is depicted as the gray-shaded region. It intersects the elliptic cylinder and confines the region of integration.
#### Axes:
- **x-axis:** Extends from \( (2,0,0) \) horizontally.
- **y-axis:** Extends from \( (2,3,0) \) vertically.
- **z-axis:** Extends from \( (0,0,4) \) upwards.
#### Intersections:
- **Intersection of Cylinder and Plane (at boundaries):**
- At \( y=0 \):
- Points: \( (2, 0, 0) \) and \( (0, 0, 4) \)
- At \( y=3 \):
- Points: \( (2, 3, 0) \) and \( (0, 3, 4) \)
- The intersection points of the elliptic cylinder with the plane \( y = 3 \) form a rectangular boundary for the region in consideration.
The diagram is useful for visualizing the limits and boundaries of the region \( D \) we are integrating over. It highlights the interaction of a three-dimensional solid constrained by both a curved surface (the elliptic cylinder) and a flat surface (the plane \(y=3\)).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F23435939-1300-49d7-9ff3-84762f7d9052%2F79c79c18-bd69-417b-b459-5df89f7ec6a1%2Fskr6d2s_processed.jpeg&w=3840&q=75)
