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 integral by changing to cylindrical coordinates:
\[
\int_{-5}^{5} \int_{0}^{\sqrt{25-x^2}} \int_{0}^{\sqrt{25-x^2-y^2}} \sqrt{x^2+y^2} \, dz \, dy \, dx
\]
**Instruction:**
Change the given Cartesian triple integral into cylindrical coordinates and evaluate it.
**Variables and Limits in Cartesian Coordinates:**
- \(x\) ranges from \(-5\) to \(5\).
- \(y\) ranges from \(0\) to \(\sqrt{25-x^2}\).
- \(z\) ranges from \(0\) to \(\sqrt{25-x^2-y^2}\).
**Conversion to Cylindrical Coordinates:**
In cylindrical coordinates:
- \(x = r \cos \theta\)
- \(y = r \sin \theta\)
- \(z = z\) (remains the same)
- The Jacobian of the transformation is \(r\).
The integral becomes:
\[
\int_{\theta_1}^{\theta_2} \int_{r_1}^{r_2} \int_{z_1}^{z_2} r \cdot \sqrt{r^2} \, dz \, dr \, d\theta
\]
**Graphical Representation:**
There are no graphs or diagrams provided in the image. The problem refers to a triple integral over a region in 3-dimensional space that is best interpreted through geometric visualization involving a quarter-cylindrical wedge extending along the z-axis within a sphere of radius \(5\).
The solution includes:
1. Determining the bounds for \(r\) and \(\theta\).
2. Evaluating the cylindrical integral using the new limits and the cylindrical transformation expressions.
This type of problem is common in multivariable calculus and involves understanding the geometry of the integration region as well as familiarity with coordinate system transformations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F507c2990-06e9-4510-adc8-e6d70ae71148%2Fe17b1204-8157-42c5-8e6b-a56edc1f20fe%2Fk6r6d4w_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Evaluate the integral by changing to cylindrical coordinates:
\[
\int_{-5}^{5} \int_{0}^{\sqrt{25-x^2}} \int_{0}^{\sqrt{25-x^2-y^2}} \sqrt{x^2+y^2} \, dz \, dy \, dx
\]
**Instruction:**
Change the given Cartesian triple integral into cylindrical coordinates and evaluate it.
**Variables and Limits in Cartesian Coordinates:**
- \(x\) ranges from \(-5\) to \(5\).
- \(y\) ranges from \(0\) to \(\sqrt{25-x^2}\).
- \(z\) ranges from \(0\) to \(\sqrt{25-x^2-y^2}\).
**Conversion to Cylindrical Coordinates:**
In cylindrical coordinates:
- \(x = r \cos \theta\)
- \(y = r \sin \theta\)
- \(z = z\) (remains the same)
- The Jacobian of the transformation is \(r\).
The integral becomes:
\[
\int_{\theta_1}^{\theta_2} \int_{r_1}^{r_2} \int_{z_1}^{z_2} r \cdot \sqrt{r^2} \, dz \, dr \, d\theta
\]
**Graphical Representation:**
There are no graphs or diagrams provided in the image. The problem refers to a triple integral over a region in 3-dimensional space that is best interpreted through geometric visualization involving a quarter-cylindrical wedge extending along the z-axis within a sphere of radius \(5\).
The solution includes:
1. Determining the bounds for \(r\) and \(\theta\).
2. Evaluating the cylindrical integral using the new limits and the cylindrical transformation expressions.
This type of problem is common in multivariable calculus and involves understanding the geometry of the integration region as well as familiarity with coordinate system transformations.
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