Find the volume of the given solid. enclosed by the paraboloid z = x² + y2 + 1 and the planes x = 0, y = 0, z = 0, and x + y = 5

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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**Problem Statement:**

Find the volume of the given solid.

The solid is enclosed by the paraboloid defined by the equation:

\[ z = x^2 + y^2 + 1 \]

and the planes:

- \( x = 0 \)
- \( y = 0 \)
- \( z = 0 \)
- \( x + y = 5 \)

**Visual Representation:**

The image contains a placeholder for a potential diagram, but no actual graph or diagram is provided. For solving this problem, one would typically create a 3D graph showing the intersections of the paraboloid and the planes to visualize the solid whose volume needs to be calculated. 

**Explanation of Key Elements:**

- **Paraboloid**: This is a three-dimensional surface described by the equation \( z = x^2 + y^2 + 1 \). It is symmetric around the z-axis and opens upwards.

- **Planes**:
  - \( x = 0 \) and \( y = 0 \) are the coordinate planes in 3D space along the y-z and x-z axes respectively.
  - \( z = 0 \) is the xy-plane.
  - \( x + y = 5 \) is a plane inclined at an angle, creating a boundary condition along the x-y plane.

To find the volume of the solid under these conditions, integration techniques in multivariable calculus would be employed, considering the bounds defined by these equations.
Transcribed Image Text:**Problem Statement:** Find the volume of the given solid. The solid is enclosed by the paraboloid defined by the equation: \[ z = x^2 + y^2 + 1 \] and the planes: - \( x = 0 \) - \( y = 0 \) - \( z = 0 \) - \( x + y = 5 \) **Visual Representation:** The image contains a placeholder for a potential diagram, but no actual graph or diagram is provided. For solving this problem, one would typically create a 3D graph showing the intersections of the paraboloid and the planes to visualize the solid whose volume needs to be calculated. **Explanation of Key Elements:** - **Paraboloid**: This is a three-dimensional surface described by the equation \( z = x^2 + y^2 + 1 \). It is symmetric around the z-axis and opens upwards. - **Planes**: - \( x = 0 \) and \( y = 0 \) are the coordinate planes in 3D space along the y-z and x-z axes respectively. - \( z = 0 \) is the xy-plane. - \( x + y = 5 \) is a plane inclined at an angle, creating a boundary condition along the x-y plane. To find the volume of the solid under these conditions, integration techniques in multivariable calculus would be employed, considering the bounds defined by these equations.
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