Let S be the solid obtained by rotating the region shown in the figure below about the y-axis. y sin(2x²) Ⓒ Graph Description Sketch the sold. Graph O What are the circumference c and heighth of a typical cylindrical shell? c(x) = Use the method of cylindrical shells to find the volume V of St Do you think this method is preferable to using washers? Explain your reasoning. Graph un umumkan -15 -1 -25 Graph Description 15 O Graph Description x Graph -15 -1 -25 25 15 O Graph Graph Description O
Let S be the solid obtained by rotating the region shown in the figure below about the y-axis. y sin(2x²) Ⓒ Graph Description Sketch the sold. Graph O What are the circumference c and heighth of a typical cylindrical shell? c(x) = Use the method of cylindrical shells to find the volume V of St Do you think this method is preferable to using washers? Explain your reasoning. Graph un umumkan -15 -1 -25 Graph Description 15 O Graph Description x Graph -15 -1 -25 25 15 O Graph Graph Description O
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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![**Volume of Solids of Revolution**
Let \( S \) be the solid obtained by rotating the region shown in the figure below about the \( y \)-axis.
### Initial Graph
The figure shows the graph of the function \( y = 6 \sin \left( \frac{\pi x}{4} \right) \) with a shaded region under the curve from \( x = 0 \) to \( x = 2 \). The shaded area represents the region that will be rotated around the \( y \)-axis to create the solid \( S \).
### Solid of Revolution
To visualize the solid, we will rotate the shaded region around the \( y \)-axis.
#### 3D Sketches
1. **First Skeletal View**
- The first 3D graph shows the full solid after the complete rotation around the \( y \)-axis. It resembles a bell or barrel shape, symmetric about the \( y \)-axis, extending from \( x = -2 \) to \( x = 2 \) and from \( y = 0 \) to \( y = 6 \).
2. **Perspective View 1**
- The second 3D graph presents a rotated view from a different angle, providing a perspective where the solid appears as an elongated shape along the \( y \)-axis.
3. **Front View**
- The third 3D graph is a front-facing view, showing a clear bell-shaped solid extending from \( y = 0 \) to \( y = 6 \).
4. **Perspective View 2**
- The fourth 3D graph shows another rotated perspective, reinforcing the previous visualizations of the solid's shape and symmetry.
### Determining Dimensions for Cylindrical Shells
To find the circumference \( c \) and height \( h \) of a typical cylindrical shell:
\[
c(x) = 2\pi x
\]
\[
h(x) = y = 6 \sin \left( \frac{\pi x}{4} \right)
\]
### Volume Calculation Using the Method of Cylindrical Shells
The volume \( V \) of the solid \( S \) can be obtained by integrating using cylindrical shells:
\[
V = \int_{0}^{2} 2\pi x \cdot 6 \sin \left( \frac{\pi x}{4} \right) \, dx](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2e97f32-5988-4deb-9547-eb6ce37eb1f3%2F99505f86-f43d-4520-b7ac-79caeace424d%2Fst1jtbs_processed.png&w=3840&q=75)
Transcribed Image Text:**Volume of Solids of Revolution**
Let \( S \) be the solid obtained by rotating the region shown in the figure below about the \( y \)-axis.
### Initial Graph
The figure shows the graph of the function \( y = 6 \sin \left( \frac{\pi x}{4} \right) \) with a shaded region under the curve from \( x = 0 \) to \( x = 2 \). The shaded area represents the region that will be rotated around the \( y \)-axis to create the solid \( S \).
### Solid of Revolution
To visualize the solid, we will rotate the shaded region around the \( y \)-axis.
#### 3D Sketches
1. **First Skeletal View**
- The first 3D graph shows the full solid after the complete rotation around the \( y \)-axis. It resembles a bell or barrel shape, symmetric about the \( y \)-axis, extending from \( x = -2 \) to \( x = 2 \) and from \( y = 0 \) to \( y = 6 \).
2. **Perspective View 1**
- The second 3D graph presents a rotated view from a different angle, providing a perspective where the solid appears as an elongated shape along the \( y \)-axis.
3. **Front View**
- The third 3D graph is a front-facing view, showing a clear bell-shaped solid extending from \( y = 0 \) to \( y = 6 \).
4. **Perspective View 2**
- The fourth 3D graph shows another rotated perspective, reinforcing the previous visualizations of the solid's shape and symmetry.
### Determining Dimensions for Cylindrical Shells
To find the circumference \( c \) and height \( h \) of a typical cylindrical shell:
\[
c(x) = 2\pi x
\]
\[
h(x) = y = 6 \sin \left( \frac{\pi x}{4} \right)
\]
### Volume Calculation Using the Method of Cylindrical Shells
The volume \( V \) of the solid \( S \) can be obtained by integrating using cylindrical shells:
\[
V = \int_{0}^{2} 2\pi x \cdot 6 \sin \left( \frac{\pi x}{4} \right) \, dx
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