Q. 26
Question
Consider the region between and the x-axis on [0, 4]. For each line of rotation given in Exercises 25–28, use four shells based on the given rectangles to approximate the volume of the resulting solid.
Step-by-Step Solution
VerifiedThe volume of solid formed is
The figure is
As the solid is rotated around the $y$-axis so the rectangles are drawn parallel to the $y$-axis, that is shells with base on the $x$-axis. On the $x$-axis the interval $[0,4]$ is divided into four subintervals viz $[0,1],[1,2],[2,3]$ and $[3,4]$, the width of each subinterval is 1 that is $\Delta x=1$. Since the region in the diagram is bounded by $f(x)=\sqrt{x}$, so height of the shells is given by $h=\sqrt{x}$
For the first shell on the interval $[0,1]$, the radius of the shell is the midpoint of $[0,1]$ that is $0.5$ also the height of the shell is found at the midpoint of $[0,1]$, that is $h=\sqrt{0.5}=0.707$
Hence volume of first shell is
$$
\begin{aligned}
2 \pi r h \Delta x &=2 \pi(0.5)(0.7071)(1) \\
&=2.221
\end{aligned}
$$
Volume of first shell is $2.221$
The second shell has the average radius equal to the midpoint of $[1,2]$ that is $1.5$ and its height computed at the midpoint is $h=\sqrt{1.5}=1.2247$
So the volume of the second shell is
$$
\begin{aligned}
2 \pi r h \Delta x &=2 \pi(1.5)(1.2247)(1) \\
&=11.543
\end{aligned}
$$
Volume of second shell is $11.543$
The third shell has the average radius equal to the midpoint of $[2,3]$ that is $2.5$ and its height computed at the midpoint is $h=\sqrt{2.5}=1.5811$
So the volume of the third shell is
$$
\begin{aligned}
2 \pi r h \Delta x &=2 \pi(2.5)(1.5811)(1) \\
&=24.836
\end{aligned}
$$
Volume of third shell is $24.836$
The fourth shell has the average radius equal to the midpoint of $[3,4]$ that is $3.5$ and its height computed at the midpoint is $h=\sqrt{3.5}=1.8708$,
So the volume of the third shell is
$$
\begin{aligned}
2 \pi r h \Delta x &=2 \pi(3.5)(1.8708)(1) \\
&=41.142
\end{aligned}
$$
Volume of fourth shell is $41.142$
Therefore, the volume of solid formed is