Answer
The volume of the darkly shaded region is \[251\text{ i}{{\text{n}}^{\text{3}}}\].
Work Step by Step
There are two cylinders in the given figure. The first cylinder is having dimension as
\[d=6\text{ inches}\] and \[h=10\text{ inches}\]. The second cylinder is having dimension as \[d=2\text{ inches}\] and \[h=10\text{ inches}\].
First of all, compute the radius of the first cylinder using the equation as shown below:
\[\begin{align}
& \text{Radius}=\frac{1}{2}\times \text{Diameter} \\
& =\frac{1}{2}\times 6\text{ inches} \\
& \text{=}3\text{ inches}
\end{align}\]
In this way, the dimensions of the first cylinder are
\[r=3\text{ inches}\] and \[h=10\text{ inches}\]
Compute the radius of the second cylinder using the equation as shown below:
\[\begin{align}
& \text{Radius}=\frac{1}{2}\times \text{Diameter} \\
& =\frac{1}{2}\times 2\text{ inches} \\
& \text{=1 inch}
\end{align}\]
In this way, the dimensions of the second cylinder are
\[r=1\text{ inch}\] and \[h=10\text{ inches}\]
Now, to compute the volume of the darkly shaded region, one is required to subtract the volume of the second cylinder from the first cylinder.
Compute the volume of the first cylinder using the equation as shown below:
\[\begin{align}
& \text{Volume of the first cylinder (}{{\text{V}}_{\text{1}}})=\pi {{r}^{2}}h \\
& \text{=}\left( \pi {{\left( 3\text{ in} \right)}^{2}}10\text{ in} \right) \\
& =\text{90}\pi \text{ i}{{\text{n}}^{\text{3}}}
\end{align}\]
Compute the volume of the first cylinder using the equation as shown below:
\[\begin{align}
& \text{Volume of the second cylinder (}{{\text{V}}_{2}})=\pi {{r}^{2}}h \\
& \text{=}\left( \pi {{\left( \text{1 in} \right)}^{2}}10\text{ in} \right) \\
& =1\text{0}\pi \text{ i}{{\text{n}}^{\text{3}}}
\end{align}\]
Now, compute the volume of the darkly shaded region using the equation as shown below:
\[\begin{align}
& \text{Volume of the darkly shaded region}=\text{Volume of first cylinder}- \\
& \text{Volume of second cylinder} \\
& =\text{90}\pi \text{ i}{{\text{n}}^{\text{3}}}-10\pi \text{ i}{{\text{n}}^{\text{3}}} \\
& =8\text{0}\pi \text{ i}{{\text{n}}^{\text{3}}} \\
& =251\text{ i}{{\text{n}}^{\text{3}}}
\end{align}\]
Hence, the volume of the darkly shaded region is \[251\text{ i}{{\text{n}}^{\text{3}}}\].
