Learning Outcomes
- Compute volumes of solids of revolution.
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.
- Length is the integral of areas.
- Area is the integral of lengths.
- Length is the derivative of areas.
- None of these.
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*}
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 \(x=0,y=4,y=x^2\text{.}\)
(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.
- Region is rotated around the \(x\)-axis; the cross-sectional area is determined by the line segment's \(x\)-value.
- Region is rotated around the \(x\)-axis; the cross-sectional area is determined by the line segment's \(y\)-value.
- Region is rotated around the \(y\)-axis; the cross-sectional area is determined by the line segment's \(x\)-value.
- Region is rotated around the \(y\)-axis; the cross-sectional area is determined by the line segment's \(y\)-value.
(b)
Which of these formulas is most appropriate to find this illustration's cross-sectional area?
- \(\displaystyle \pi r^2\)
- \(\displaystyle 2\pi rh\)
- \(\displaystyle \pi R^2-\pi r^2\)
- \(\displaystyle \frac{1}{2}bh\)
(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.
- Region is rotated around the \(x\)-axis; the cross-sectional area is determined by the line segment's \(x\)-value.
- Region is rotated around the \(x\)-axis; the cross-sectional area is determined by the line segment's \(y\)-value.
- Region is rotated around the \(y\)-axis; the cross-sectional area is determined by the line segment's \(x\)-value.
- Region is rotated around the \(y\)-axis; the cross-sectional area is determined by the line segment's \(y\)-value.
(d)
Which of these formulas is most appropriate to find this illustration's cross-sectional area?
- \(\displaystyle \pi r^2\)
- \(\displaystyle 2\pi rh\)
- \(\displaystyle \pi R^2-\pi r^2\)
- \(\displaystyle \frac{1}{2}bh\)
(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.
- Region is rotated around the \(x\)-axis; the cross-sectional area is determined by the line segment's \(x\)-value.
- Region is rotated around the \(x\)-axis; the cross-sectional area is determined by the line segment's \(y\)-value.
- Region is rotated around the \(y\)-axis; the cross-sectional area is determined by the line segment's \(x\)-value.
- Region is rotated around the \(y\)-axis; the cross-sectional area is determined by the line segment's \(y\)-value.
(f)
Which of these formulas is most appropriate to find this illustration's cross-sectional area?
- \(\displaystyle \pi r^2\)
- \(\displaystyle 2\pi rh\)
- \(\displaystyle \pi R^2-\pi r^2\)
- \(\displaystyle \frac{1}{2}bh\)
(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.
- Region is rotated around the \(x\)-axis; the cross-sectional area is determined by the line segment's \(x\)-value.
- Region is rotated around the \(x\)-axis; the cross-sectional area is determined by the line segment's \(y\)-value.
- Region is rotated around the \(y\)-axis; the cross-sectional area is determined by the line segment's \(x\)-value.
- Region is rotated around the \(y\)-axis; the cross-sectional area is determined by the line segment's \(y\)-value.
(h)
Which of these formulas is most appropriate to find this illustration's cross-sectional area?
- \(\displaystyle \pi r^2\)
- \(\displaystyle 2\pi rh\)
- \(\displaystyle \pi R^2-\pi r^2\)
- \(\displaystyle \frac{1}{2}bh\)
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.
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*}
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*}
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*}
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.
(a)
Sketch two copies of this region in the \(xy\) plane.
(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.
(c)
Consider the method required for each cross-section drawn. Which would be the easiest strategy to proceed with?
- The horizontal line segment, using the disk/washer method.
- The horizontal line segment, using the shell method.
- The vertical line segment, using the disk/washer method.
- The vertical line segment, using the shell method.
(d)
Let's proceed with the vertical segment. Which formula is most appropriate for the radius?
- \(\displaystyle r(x)=x\)
- \(\displaystyle r(x)=5e^x\)
- \(\displaystyle r(x)=5\ln(x)\)
- \(\displaystyle r(x)=\frac{1}{5}\ln(x)\)
(e)
Which of these integrals is equal to the volume of the solid of revolution?
- \(\displaystyle \int_0^1 25\pi e^{2x}\,dx\)
- \(\displaystyle \int_0^1 5\pi^2 e^{x}\,dx\)
- \(\displaystyle \int_0^2 25\pi e^{x}\,dx\)
- \(\displaystyle \int_0^2 5\pi^2 e^{2x}\,dx\)
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.
(a)
Sketch two copies of this region in the \(xy\) plane.
(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.
(c)
Consider the method required for each cross-section drawn. Which would be the easiest strategy to proceed with?
- The horizontal line segment, using the disk/washer method.
- The horizontal line segment, using the shell method.
- The vertical line segment, using the disk/washer method.
- The vertical line segment, using the shell method.
(d)
Let's proceed with the vertical segment. Which formula is most appropriate for the radius?
- \(\displaystyle r(x)=x\)
- \(\displaystyle r(x)=5e^x\)
- \(\displaystyle r(x)=5\ln(x)\)
- \(\displaystyle r(x)=\frac{1}{5}\ln(x)\)
(e)
Which formula is most appropriate for the height?
- \(\displaystyle h(x)=x\)
- \(\displaystyle h(x)=5e^x\)
- \(\displaystyle h(x)=5\ln(x)\)
- \(\displaystyle h(x)=\frac{1}{5}\ln(x)\)
(f)
Which of these integrals is equal to the volume of the solid of revolution?
- \(\displaystyle \int_0^1 5\pi^2 xe^{x}\,dx\)
- \(\displaystyle \int_0^1 10\pi xe^{x}\,dx\)
- \(\displaystyle \int_0^2 5\pi xe^{x}\,dx\)
- \(\displaystyle \int_0^2 10\pi x^2e^{x}\,dx\)
Activity 6.3.7.
Consider the region bounded by \(y=2 \, x + 3, y=0, x=4, x=7\text{.}\)
(a)
Find an integral which computes the volume of the solid formed by rotating this region about the \(x\)-axis.
(b)
Find an integral which computes the volume of the solid formed by rotating this region about the \(y\)-axis.
