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Calculus for Team-Based Inquiry Learning PREVIEW Edition

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Section 6.3 Volumes of Revolution (AI3)

Subsection 6.3.1 Activities

Activity 6.3.1.

Consider the following visualization to decide which of these statements is most appropriate for describing the relationship of lengths and areas.
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Diagram Exploration Keyboard Controls
Key Action
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B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
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  1. Length is the integral of areas.
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  2. Area is the integral of lengths.
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  3. Length is the derivative of areas.
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  4. None of these.
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Definition 6.3.2.

We define the volume of a solid with cross sectional area given by \(A(x)\) laying between \(a\leq x\leq b\) to be the definite integral
\begin{equation*} \mathrm{Volume}=\int_a^b A(x)\,dx\text{.} \end{equation*}
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A three dimensional interactive showing a (half of a) circular cone, with a circular cross section.
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Activity 6.3.3.

We will be focused on the volumes of solids obtained by revolving a region around an axis. Let’s use the running example of the region bounded by the curves \(x=0,y=4,y=x^2\text{.}\)
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Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
(a)
Consider the below illustrated revolution of this region, and the cross-section drawn from a horizontal line segment. Choose the most appropriate description of this illustration.
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A 3D interactive showing the solid region formed by rotating the region around the \(y\)-axis. The parabola \(y=x^2\) is highlighted in red along the outside of the solid. A circular cross section is shown, with its radius also drawn in red, extending from the \(y\)-axis to the parabola.
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  1. Region is rotated around the \(x\)-axis; the cross-sectional area is determined by the line segment’s \(x\)-value.
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  2. Region is rotated around the \(x\)-axis; the cross-sectional area is determined by the line segment’s \(y\)-value.
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  3. Region is rotated around the \(y\)-axis; the cross-sectional area is determined by the line segment’s \(x\)-value.
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  4. Region is rotated around the \(y\)-axis; the cross-sectional area is determined by the line segment’s \(y\)-value.
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(b)
Which of these formulas is most appropriate to find this illustration’s cross-sectional area?
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  1. \(\displaystyle \pi r^2\)
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  2. \(\displaystyle 2\pi rh\)
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  3. \(\displaystyle \pi R^2-\pi r^2\)
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  4. \(\displaystyle \frac{1}{2}bh\)
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(c)
Consider the below illustrated revolution of this region, and the cross-section drawn from a vertical line segment. Choose the most appropriate description of this illustration.
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A 3D interactive showing the solid region formed by rotating the region around the \(y\)-axis. The parabola \(y=x^2\) is highlighted in red along the outside of the solid. A cylindrical cross section parallel to the \(y\)-axis is shownis shown, with its height also drawn in red, extending from the parabola to the line \(y=4\text{.}\)
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  1. Region is rotated around the \(x\)-axis; the cross-sectional area is determined by the line segment’s \(x\)-value.
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  2. Region is rotated around the \(x\)-axis; the cross-sectional area is determined by the line segment’s \(y\)-value.
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  3. Region is rotated around the \(y\)-axis; the cross-sectional area is determined by the line segment’s \(x\)-value.
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  4. Region is rotated around the \(y\)-axis; the cross-sectional area is determined by the line segment’s \(y\)-value.
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(d)
Which of these formulas is most appropriate to find this illustration’s cross-sectional area?
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  1. \(\displaystyle \pi r^2\)
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  2. \(\displaystyle 2\pi rh\)
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  3. \(\displaystyle \pi R^2-\pi r^2\)
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  4. \(\displaystyle \frac{1}{2}bh\)
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(e)
Consider the below illustrated revolution of this region, and the cross-section drawn from a horizontal line segment. Choose the most appropriate description of this illustration.
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A 3D interactive showing the solid region formed by rotating the region around the \(x\)-axis. The parabola \(y=x^2\) is highlighted in red along the outside of the solid. A cylindrical cross section parallel to the \(x\)-axis is shown, with its height also drawn in red, extending from the \(y\)-axis to the parabola.
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  1. Region is rotated around the \(x\)-axis; the cross-sectional area is determined by the line segment’s \(x\)-value.
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  2. Region is rotated around the \(x\)-axis; the cross-sectional area is determined by the line segment’s \(y\)-value.
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  3. Region is rotated around the \(y\)-axis; the cross-sectional area is determined by the line segment’s \(x\)-value.
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  4. Region is rotated around the \(y\)-axis; the cross-sectional area is determined by the line segment’s \(y\)-value.
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(f)
Which of these formulas is most appropriate to find this illustration’s cross-sectional area?
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  1. \(\displaystyle \pi r^2\)
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  2. \(\displaystyle 2\pi rh\)
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  3. \(\displaystyle \pi R^2-\pi r^2\)
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  4. \(\displaystyle \frac{1}{2}bh\)
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(g)
Consider the below illustrated revolution of this region, and the cross-section drawn from a vertical line segment. Choose the most appropriate description of this illustration.
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A 3D interactive showing the solid region formed by rotating the region around the \(x\)-axis. The parabola \(y=x^2\) is highlighted in red along the outside of the solid. An annular cross section orthogonal to the \(x\)-axis is shown, with its radial difference also drawn in red, extending from the parabola to the line \(y=4\text{.}\)
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  1. Region is rotated around the \(x\)-axis; the cross-sectional area is determined by the line segment’s \(x\)-value.
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  2. Region is rotated around the \(x\)-axis; the cross-sectional area is determined by the line segment’s \(y\)-value.
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  3. Region is rotated around the \(y\)-axis; the cross-sectional area is determined by the line segment’s \(x\)-value.
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  4. Region is rotated around the \(y\)-axis; the cross-sectional area is determined by the line segment’s \(y\)-value.
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(h)
Which of these formulas is most appropriate to find this illustration’s cross-sectional area?
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  1. \(\displaystyle \pi r^2\)
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  2. \(\displaystyle 2\pi rh\)
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  3. \(\displaystyle \pi R^2-\pi r^2\)
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  4. \(\displaystyle \frac{1}{2}bh\)
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Remark 6.3.4.

Generally when solving problems without the aid of technology, it’s useful to draw your region in two dimensions, choose whether to use a horizontal or vertical line segment, and draw its rotation to determine the cross-sectional shape.
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When the shape is a disk, this is called the disk method and we use one of these formulas depending on whether the cross-sectional area depends on \(x\) or \(y\text{.}\)
\begin{equation*} V=\int_a^b \pi r(x)^2\,dx,\hspace{2em}V=\int_a^b \pi r(y)^2\,dy\text{.} \end{equation*}
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When the shape is a washer, this is called the washer method and we use one of these formulas depending on whether the cross-sectional area depends on \(x\) or \(y\text{.}\)
\begin{equation*} V=\int_a^b\left(\pi R(x)^2- \pi r(x)^2\right)\,dx,\hspace{2em}V=\int_a^b\left(\pi R(y)^2- \pi r(y)^2\right)\,dy\text{.} \end{equation*}
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When the shape is a cylindrical shell, this is called the shell method and we use one of these formulas depending on whether the cross-sectional area depends on \(x\) or \(y\text{.}\)
\begin{equation*} V=\int_a^b 2\pi r(x)h(x)\,dx,\hspace{2em}V=\int_a^b 2\pi r(y)h(y)\,dy\text{.} \end{equation*}
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Activity 6.3.5.

Let’s now consider the region bounded by the curves \(x=0,x=1,y=0,y=5e^x\text{,}\) rotated about the \(x\)-axis.
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(a)
Sketch two copies of this region in the \(xy\) plane.
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(b)
Draw a vertical line segment in one region and its rotation around the \(x\)-axis. Draw a horizontal line segment in the other region and its rotation around the \(x\)-axis.
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(c)
Consider the method required for each cross-section drawn. Which would be the easiest strategy to proceed with?
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  1. The horizontal line segment, using the disk/washer method.
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  2. The horizontal line segment, using the shell method.
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  3. The vertical line segment, using the disk/washer method.
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  4. The vertical line segment, using the shell method.
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(d)
Let’s proceed with the vertical segment. Which formula is most appropriate for the radius?
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  1. \(\displaystyle r(x)=x\)
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  2. \(\displaystyle r(x)=5e^x\)
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  3. \(\displaystyle r(x)=5\ln(x)\)
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  4. \(\displaystyle r(x)=\frac{1}{5}\ln(x)\)
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(e)
Which of these integrals is equal to the volume of the solid of revolution?
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  1. \(\displaystyle \int_0^1 25\pi e^{2x}\,dx\)
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  2. \(\displaystyle \int_0^1 5\pi^2 e^{x}\,dx\)
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  3. \(\displaystyle \int_0^2 25\pi e^{x}\,dx\)
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  4. \(\displaystyle \int_0^2 5\pi^2 e^{2x}\,dx\)
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Activity 6.3.6.

Let’s now consider the same region, bounded by the curves \(x=0,x=1,y=0,y=5e^x\text{,}\) but this time rotated about the \(y\)-axis.
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(a)
Sketch two copies of this region in the \(xy\) plane.
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(b)
Draw a vertical line segment in one region and its rotation around the \(y\)-axis. Draw a horizontal line segment in the other region and its rotation around the \(y\)-axis.
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(c)
Consider the method required for each cross-section drawn. Which would be the easiest strategy to proceed with?
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  1. The horizontal line segment, using the disk/washer method.
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  2. The horizontal line segment, using the shell method.
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  3. The vertical line segment, using the disk/washer method.
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  4. The vertical line segment, using the shell method.
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(d)
Let’s proceed with the vertical segment. Which formula is most appropriate for the radius?
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  1. \(\displaystyle r(x)=x\)
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  2. \(\displaystyle r(x)=5e^x\)
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  3. \(\displaystyle r(x)=5\ln(x)\)
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  4. \(\displaystyle r(x)=\frac{1}{5}\ln(x)\)
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(e)
Which formula is most appropriate for the height?
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  1. \(\displaystyle h(x)=x\)
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  2. \(\displaystyle h(x)=5e^x\)
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  3. \(\displaystyle h(x)=5\ln(x)\)
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  4. \(\displaystyle h(x)=\frac{1}{5}\ln(x)\)
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(f)
Which of these integrals is equal to the volume of the solid of revolution?
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  1. \(\displaystyle \int_0^1 5\pi^2 xe^{x}\,dx\)
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  2. \(\displaystyle \int_0^1 10\pi xe^{x}\,dx\)
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  3. \(\displaystyle \int_0^2 5\pi xe^{x}\,dx\)
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  4. \(\displaystyle \int_0^2 10\pi x^2e^{x}\,dx\)
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Activity 6.3.7.

Consider the region bounded by \(y=2 \, x + 3, y=0, x=4, x=7\text{.}\)
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(a)
Find an integral which computes the volume of the solid formed by rotating this region about the \(x\)-axis.
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(b)
Find an integral which computes the volume of the solid formed by rotating this region about the \(y\)-axis.
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Subsection 6.3.2 Videos

Figure 89. Video: Compute volumes of solids of revolution, washer \(x\)-axis
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Figure 90. Video: Compute volumes of solids of revolution, shell \(x\)-axis
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Figure 91. Video: Compute volumes of solids of revolution, washer \(y\)-axis
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Figure 92. Video: Compute volumes of solids of revolution, shell \(y\)-axis
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Subsection 6.3.3 Exercises

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