1. Sketch the solid whose volume is given by the iterated integral. 3/2 (-) fr r dz dr do
1. Sketch the solid whose volume is given by the iterated integral. 3/2 (-) fr r dz dr do
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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![**Iterated Integrals and Volumes of Solids**
This section focuses on sketching the solid whose volume is defined by the given iterated integrals. Each of these integrals represents the volume of a solid in cylindrical coordinates.
### Problem Statement:
**4. Sketch the solid whose volume is given by the iterated integral.**
**(a)**
\[
\int_{\pi/2}^{3\pi/2} \int_{0}^{4} \int_{-1}^{2} r \, dz \, dr \, d\theta
\]
**(b)**
\[
\int_{-\pi/2}^{\pi/2} \int_{0}^{2} \int_{r}^{r^2} r \, dz \, dr \, d\theta
\]
**(c)**
\[
\int_{0}^{2} \int_{0}^{2\pi} \int_{0}^{r} r \, dz \, d\theta \, dr
\]
### Explanation of the Integrals
#### a) Integral (a):
\[
\int_{\pi/2}^{3\pi/2} \int_{0}^{4} \int_{-1}^{2} r \, dz \, dr \, d\theta
\]
- \(\theta\) ranges from \(\pi/2\) to \(3\pi/2\), which describes half of a full revolution around the z-axis.
- \(r\) ranges from 0 to 4, indicating the radial distance from the z-axis.
- \(z\) ranges from -1 to 2, covering vertical positions.
This integral represents a cylindrical segment.
#### b) Integral (b):
\[
\int_{-\pi/2}^{\pi/2} \int_{0}^{2} \int_{r}^{r^2} r \, dz \, dr \, d\theta
\]
- \(\theta\) ranges from \(-\pi/2\) to \(\pi/2\), representing a half revolution.
- \(r\) ranges from 0 to 2, indicating radial distance.
- \(z\) ranges from \(r\) to \(r^2\), suggesting that the limits for z depend on the value of \(r\), adding a complexity indicating that height varies with radius.
This integral](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcbfb1871-07e2-4dce-8005-269462079c4a%2F7ec76d89-6bd2-4b0f-940c-9be3c32a69c4%2F5jocrv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Iterated Integrals and Volumes of Solids**
This section focuses on sketching the solid whose volume is defined by the given iterated integrals. Each of these integrals represents the volume of a solid in cylindrical coordinates.
### Problem Statement:
**4. Sketch the solid whose volume is given by the iterated integral.**
**(a)**
\[
\int_{\pi/2}^{3\pi/2} \int_{0}^{4} \int_{-1}^{2} r \, dz \, dr \, d\theta
\]
**(b)**
\[
\int_{-\pi/2}^{\pi/2} \int_{0}^{2} \int_{r}^{r^2} r \, dz \, dr \, d\theta
\]
**(c)**
\[
\int_{0}^{2} \int_{0}^{2\pi} \int_{0}^{r} r \, dz \, d\theta \, dr
\]
### Explanation of the Integrals
#### a) Integral (a):
\[
\int_{\pi/2}^{3\pi/2} \int_{0}^{4} \int_{-1}^{2} r \, dz \, dr \, d\theta
\]
- \(\theta\) ranges from \(\pi/2\) to \(3\pi/2\), which describes half of a full revolution around the z-axis.
- \(r\) ranges from 0 to 4, indicating the radial distance from the z-axis.
- \(z\) ranges from -1 to 2, covering vertical positions.
This integral represents a cylindrical segment.
#### b) Integral (b):
\[
\int_{-\pi/2}^{\pi/2} \int_{0}^{2} \int_{r}^{r^2} r \, dz \, dr \, d\theta
\]
- \(\theta\) ranges from \(-\pi/2\) to \(\pi/2\), representing a half revolution.
- \(r\) ranges from 0 to 2, indicating radial distance.
- \(z\) ranges from \(r\) to \(r^2\), suggesting that the limits for z depend on the value of \(r\), adding a complexity indicating that height varies with radius.
This integral
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