Consider the region between the graph of $f ( x ) = \sqrt { x - 1 }$ and the x-axis on the interval [1, 5].
-Using four shells approximate the volume of the solid that is obtained by revolving this region around the x-axis.
A) 9.25 $\pi$
B) 8.25 $\pi$
C) 10.25 $\pi$
D) 7.25 $\pi$

Question

Quizplus · Accepted Answer

The volume of the solid formed by revolving the region between the graph of $f(x) = \sqrt{x - 1}$ and the x-axis from $x = 1$ to $x = 5$ around the x-axis can be approximated using the method of cylindrical shells. The formula for the volume of a single shell is $V = 2\pi rh$, where $r$ is the average radius of the shell and $h$ is the height of the shell.Dividing the interval [1, 5] into four parts, we get the points $x = 1, 2, 3, 4, 5$. The thickness of each shell ($\Delta x$) is $1$.For each shell:- At $x = 1$, $r = 0$ and $h = 0$, so the volume is $0$.- At $x = 2$, $r = 1.5$ (average of 1 and 2) and $h = \sqrt{2 - 1} = 1$, so the volume is $2\pi(1.5)(1) = 3\pi$.- At $x = 3$, $r = 2.5$ and $h = \sqrt{3 - 1} = \sqrt{2}$, so the volume is $2\pi(2.5)(\sqrt{2})$.- At $x = 4$, $r = 3.5$ and $h = \sqrt{4 - 1} = \sqrt{3}$, so the volume is $2\pi(3.5)(\sqrt{3})$.- At $x = 5$, $r = 4.5$ and $h = \sqrt{5 - 1} = 2$, so the volume is $2\pi(4.5)(2) = 18\pi$.However, the correct approach to approximate the volume using the method of shells involves calculating the volume of each shell and summing them up. The explanation above incorrectly calculates the volumes for each shell, leading to confusion. The correct approach is to use the formula for the volume of a shell, $V = 2\pi \int_{a}^{b} x f(x) dx$, and approximate this integral using the midpoint rule or another numerical method with the given intervals.The correct approximation using shells for each segment with midpoint values should be calculated properly. However, the provided explanation does not correctly follow through with the calculation for each shell, leading to an incorrect explanation of how to arrive at the correct answer. The correct answer involves calculating the volume of each shell correctly and summing them, which should approximate to $8.25\pi$, considering the correct method and calculations for the cylindrical shells method.

Consider the Region Between the Graph Of f(x)=x−1f ( x ) = \sqrt { x - 1 }f(x)=x−1​

Consider the Region Between the Graph Of $f ( x ) = \sqrt { x - 1 }$