Sketch the solid whose volume is given by the iterated integral. (8 - x - y)dy dx (0, 0, 8) (0, 0, 1) (0, 1,0) (0, 8, 0) (1, 0, 0). (8, 0, 0) (0, 0, 8) A(0, 0, 1) (0, 8, 0) (0, 1, 0) (8, 0, 0) (1, 0, 0)
Sketch the solid whose volume is given by the iterated integral. (8 - x - y)dy dx (0, 0, 8) (0, 0, 1) (0, 1,0) (0, 8, 0) (1, 0, 0). (8, 0, 0) (0, 0, 8) A(0, 0, 1) (0, 8, 0) (0, 1, 0) (8, 0, 0) (1, 0, 0)
Calculus: Early Transcendentals
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Author:James Stewart
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Chapter1: Functions And Models
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![**Sketching the Solid Defined by an Iterated Integral**
Consider the task of sketching the solid whose volume is described by the following iterated integral:
\[
\int_{0}^{8} \int_{0}^{8-x} (8-x-y) \, dy \, dx
\]
### Description of the Diagrams
There are two main diagrams, each with two indices, which illustrate the three-dimensional solid from different perspectives. These diagrams are essential for understanding how the volume is enclosed.
**First Diagram (Top-Left):**
- The coordinate axes are labeled \( x \), \( y \), and \( z \).
- The vertices of the solid are at points \( (0, 0, 8) \), \( (1, 0, 0) \), and \( (0, 1, 0) \).
- It presents a triangular face on the \( yz \)-plane.
- The solid is bounded at the top by the plane \( z = 8 - x - y \).
**Second Diagram (Top-Right):**
- The orientation of the axes remains consistent.
- Vertices are located at \( (0, 0, 1) \), \( (8, 0, 0) \), and \( (0, 8, 0) \).
- This shows a different triangular face, providing another spatial perspective.
- The bounding plane remains \( z = 8 - x - y \).
**Third Diagram (Bottom-Left):**
- Parallel to the first diagram with similar orientation.
- Vertices are at \( (0, 0, 8) \), \( (8, 0, 0) \), and \( (0, 8, 0) \).
- Displays how the plane intersects the \( xz \)-plane.
**Fourth Diagram (Bottom-Right):**
- Similarly oriented as the second diagram.
- Points are \( (0, 0, 1) \), \( (1, 0, 0) \), and \( (0, 1, 0) \).
- Offers another perspective on the spatial volume enclosed by the solid.
### Interpretation
These diagrams collectively depict the solid as a 3D shape bounded by the planes and point constraints given by the iterated integral. Each shows how different planes intersect at defined points to form the volume described by the integral. Through these views, one can](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f8196ab-d049-4153-8da6-ad35b5ef5f14%2F6f2fd539-ffe4-4240-aa71-826a6b3ec5fc%2Fzq9epk6_processed.png&w=3840&q=75)
Transcribed Image Text:**Sketching the Solid Defined by an Iterated Integral**
Consider the task of sketching the solid whose volume is described by the following iterated integral:
\[
\int_{0}^{8} \int_{0}^{8-x} (8-x-y) \, dy \, dx
\]
### Description of the Diagrams
There are two main diagrams, each with two indices, which illustrate the three-dimensional solid from different perspectives. These diagrams are essential for understanding how the volume is enclosed.
**First Diagram (Top-Left):**
- The coordinate axes are labeled \( x \), \( y \), and \( z \).
- The vertices of the solid are at points \( (0, 0, 8) \), \( (1, 0, 0) \), and \( (0, 1, 0) \).
- It presents a triangular face on the \( yz \)-plane.
- The solid is bounded at the top by the plane \( z = 8 - x - y \).
**Second Diagram (Top-Right):**
- The orientation of the axes remains consistent.
- Vertices are located at \( (0, 0, 1) \), \( (8, 0, 0) \), and \( (0, 8, 0) \).
- This shows a different triangular face, providing another spatial perspective.
- The bounding plane remains \( z = 8 - x - y \).
**Third Diagram (Bottom-Left):**
- Parallel to the first diagram with similar orientation.
- Vertices are at \( (0, 0, 8) \), \( (8, 0, 0) \), and \( (0, 8, 0) \).
- Displays how the plane intersects the \( xz \)-plane.
**Fourth Diagram (Bottom-Right):**
- Similarly oriented as the second diagram.
- Points are \( (0, 0, 1) \), \( (1, 0, 0) \), and \( (0, 1, 0) \).
- Offers another perspective on the spatial volume enclosed by the solid.
### Interpretation
These diagrams collectively depict the solid as a 3D shape bounded by the planes and point constraints given by the iterated integral. Each shows how different planes intersect at defined points to form the volume described by the integral. Through these views, one can
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