Use a triple integral to find the volume of the solid bounded by the graphs of the equations. z = xy, z = 0, 0≤x≤ 3, 0≤ y ≤ 2
Use a triple integral to find the volume of the solid bounded by the graphs of the equations. z = xy, z = 0, 0≤x≤ 3, 0≤ y ≤ 2
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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![**Triple Integral to Find Volume**
To determine the volume of the solid bounded by the given equations using a triple integral, consider the following boundaries:
- **Surface Equation**: \( z = xy \)
- **Plane at Zero Height**: \( z = 0 \)
- **Boundaries for \( x \)**: \( 0 \leq x \leq 3 \)
- **Boundaries for \( y \)**: \( 0 \leq y \leq 2 \)
**Explanation of Equations:**
1. **Surface**: \( z = xy \)
- This is a surface in three-dimensional space where the height \( z \) at any point \((x, y)\) is given by the product of \( x \) and \( y \).
2. **Plane at Zero Height**: \( z = 0 \)
- This represents the base plane (or xy-plane) where the height \( z \) is zero.
3. **Boundaries for \( x \) and \( y \)**:
- \( x \) ranges from 0 to 3.
- \( y \) ranges from 0 to 2.
**Steps to Solve Using a Triple Integral:**
1. **Set up the Triple Integral**:
\[
\int_{x=0}^{3} \int_{y=0}^{2} \int_{z=0}^{xy} dz \, dy \, dx
\]
2. **Evaluate the Inner Integral**: Perform the integration with respect to \( z \).
3. **Evaluate the Middle Integral**: Perform the integration with respect to \( y \).
4. **Evaluate the Outer Integral**: Perform the integration with respect to \( x \).
**Objective**: Calculating this integral will yield the volume of the solid bounded by these surfaces and planes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7e91f5e6-ad6f-436a-b13b-38734a3209ad%2Fcb093fe7-2acc-4b3e-b711-0b1e7da4a435%2Fkpod8a_processed.png&w=3840&q=75)
Transcribed Image Text:**Triple Integral to Find Volume**
To determine the volume of the solid bounded by the given equations using a triple integral, consider the following boundaries:
- **Surface Equation**: \( z = xy \)
- **Plane at Zero Height**: \( z = 0 \)
- **Boundaries for \( x \)**: \( 0 \leq x \leq 3 \)
- **Boundaries for \( y \)**: \( 0 \leq y \leq 2 \)
**Explanation of Equations:**
1. **Surface**: \( z = xy \)
- This is a surface in three-dimensional space where the height \( z \) at any point \((x, y)\) is given by the product of \( x \) and \( y \).
2. **Plane at Zero Height**: \( z = 0 \)
- This represents the base plane (or xy-plane) where the height \( z \) is zero.
3. **Boundaries for \( x \) and \( y \)**:
- \( x \) ranges from 0 to 3.
- \( y \) ranges from 0 to 2.
**Steps to Solve Using a Triple Integral:**
1. **Set up the Triple Integral**:
\[
\int_{x=0}^{3} \int_{y=0}^{2} \int_{z=0}^{xy} dz \, dy \, dx
\]
2. **Evaluate the Inner Integral**: Perform the integration with respect to \( z \).
3. **Evaluate the Middle Integral**: Perform the integration with respect to \( y \).
4. **Evaluate the Outer Integral**: Perform the integration with respect to \( x \).
**Objective**: Calculating this integral will yield the volume of the solid bounded by these surfaces and planes.
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