Use a triple integral to find the volume of the solid bounded by the paraboloids z 3.x² + 3y? and z = 4 – a² – y². -
Use a triple integral to find the volume of the solid bounded by the paraboloids z 3.x² + 3y? and z = 4 – a² – y². -
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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![**Topic: Calculating Volume Using Triple Integrals**
**Problem:**
Use a triple integral to find the volume of the solid bounded by the paraboloids \( z = 3x^2 + 3y^2 \) and \( z = 4 - x^2 - y^2 \).
**Explanation:**
This problem involves determining the volume of a three-dimensional region bound by two paraboloids. These surfaces are described by the given equations and can be visualized as shapes curving upwards and downwards, respectively.
To find the volume between these two surfaces, you'll set up a triple integral in cylindrical or Cartesian coordinates, where the limits of integration are determined by the points of intersection of the paraboloids.
**Steps:**
1. **Find the Intersection Curve:**
Equate the equations of the paraboloids to find the region in the \( xy \)-plane that bounds the volume:
\[
3x^2 + 3y^2 = 4 - x^2 - y^2
\]
2. **Simplify the Equation:**
Combine like terms and simplify to find the boundary in \( xy \)-plane.
3. **Set Up the Triple Integral:**
Determine appropriate limits of integration for \( x \), \( y \), and \( z \) based on the region of integration found.
4. **Evaluate the Integral:**
Compute the integral to find the volume of the solid. The difference of the functions will form the integrand, integrating over the region in the \( xy \)-plane.
Through these steps, you will obtain the volume enclosed between the two surfaces.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F763773a4-b59e-428b-8a3f-bc3dd5fb97d6%2Ff071c71b-6988-481d-a350-eed21f2f5c2e%2Fg41caer_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Calculating Volume Using Triple Integrals**
**Problem:**
Use a triple integral to find the volume of the solid bounded by the paraboloids \( z = 3x^2 + 3y^2 \) and \( z = 4 - x^2 - y^2 \).
**Explanation:**
This problem involves determining the volume of a three-dimensional region bound by two paraboloids. These surfaces are described by the given equations and can be visualized as shapes curving upwards and downwards, respectively.
To find the volume between these two surfaces, you'll set up a triple integral in cylindrical or Cartesian coordinates, where the limits of integration are determined by the points of intersection of the paraboloids.
**Steps:**
1. **Find the Intersection Curve:**
Equate the equations of the paraboloids to find the region in the \( xy \)-plane that bounds the volume:
\[
3x^2 + 3y^2 = 4 - x^2 - y^2
\]
2. **Simplify the Equation:**
Combine like terms and simplify to find the boundary in \( xy \)-plane.
3. **Set Up the Triple Integral:**
Determine appropriate limits of integration for \( x \), \( y \), and \( z \) based on the region of integration found.
4. **Evaluate the Integral:**
Compute the integral to find the volume of the solid. The difference of the functions will form the integrand, integrating over the region in the \( xy \)-plane.
Through these steps, you will obtain the volume enclosed between the two surfaces.
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