Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem Statement**
Evaluate the triple integral
\[
\iiint_E e^z \, dV
\]
where \( E \) is enclosed by the following surfaces:
- The paraboloid defined by \( z = 5 + x^2 + y^2 \).
- The cylinder given by \( x^2 + y^2 = 5 \).
- The \( xy \)-plane.
**Explanation of Concepts**
This is a problem of calculating the volume integral of the function \( e^z \) over a specific region \( E \). The region is bounded by three surfaces:
1. **Paraboloid:** The equation \( z = 5 + x^2 + y^2 \) describes a paraboloid opening upwards. This surface suggests that for a given point \((x, y)\), the height \( z \) is determined by a base value of 5 plus the square of the distance from the origin in the \( xy \)-plane.
2. **Cylinder:** The equation \( x^2 + y^2 = 5 \) denotes a vertical cylinder centered along the z-axis with a radius\(\sqrt{5}\). This imposes a radial boundary for the integration.
3. **xy-plane:** This plane is simply the \( z = 0 \) surface, serving as the lower boundary for the region of integration.
**Approach to Solution**
To evaluate the integral, consider transforming to cylindrical coordinates because of the symmetry introduced by the cylindrical boundary. In cylindrical coordinates, the relationships are:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)
- \( z = z \)
The differential volume element \( dV \) in cylindrical coordinates is expressed as \( r \, dr \, d\theta \, dz \).
**Bounds for the Integration:**
- \( r \) ranges from 0 to \(\sqrt{5}\) (because of the cylinder \( x^2 + y^2 = 5 \)).
- \( \theta \) ranges from 0 to \( 2\pi \) (a full rotation around the z-axis).
- \( z \) ranges from 0 (the \( xy \)-plane) to \( 5 + r^2 \) (since \( z \) is bounded above by the paraboloid).
This setup allows you to proceed with evaluating the integral within these](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2ac4854-06b1-4431-8395-fa3393246211%2F54aea4d3-5b5a-4ab0-bfa6-93f0d67eda9f%2Fajta8xe_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Evaluate the triple integral
\[
\iiint_E e^z \, dV
\]
where \( E \) is enclosed by the following surfaces:
- The paraboloid defined by \( z = 5 + x^2 + y^2 \).
- The cylinder given by \( x^2 + y^2 = 5 \).
- The \( xy \)-plane.
**Explanation of Concepts**
This is a problem of calculating the volume integral of the function \( e^z \) over a specific region \( E \). The region is bounded by three surfaces:
1. **Paraboloid:** The equation \( z = 5 + x^2 + y^2 \) describes a paraboloid opening upwards. This surface suggests that for a given point \((x, y)\), the height \( z \) is determined by a base value of 5 plus the square of the distance from the origin in the \( xy \)-plane.
2. **Cylinder:** The equation \( x^2 + y^2 = 5 \) denotes a vertical cylinder centered along the z-axis with a radius\(\sqrt{5}\). This imposes a radial boundary for the integration.
3. **xy-plane:** This plane is simply the \( z = 0 \) surface, serving as the lower boundary for the region of integration.
**Approach to Solution**
To evaluate the integral, consider transforming to cylindrical coordinates because of the symmetry introduced by the cylindrical boundary. In cylindrical coordinates, the relationships are:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)
- \( z = z \)
The differential volume element \( dV \) in cylindrical coordinates is expressed as \( r \, dr \, d\theta \, dz \).
**Bounds for the Integration:**
- \( r \) ranges from 0 to \(\sqrt{5}\) (because of the cylinder \( x^2 + y^2 = 5 \)).
- \( \theta \) ranges from 0 to \( 2\pi \) (a full rotation around the z-axis).
- \( z \) ranges from 0 (the \( xy \)-plane) to \( 5 + r^2 \) (since \( z \) is bounded above by the paraboloid).
This setup allows you to proceed with evaluating the integral within these
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