Answer:
The volume of the solid is 714.887 units³
Step-by-step explanation:
* Lets talk about the shell method
- The shell method is to finding the volume by decomposing
a solid of revolution into cylindrical shells
- Consider a region in the plane that is divided into thin vertical
rectangle
- If each vertical rectangle is revolved about the y-axis, we
obtain a cylindrical shell, with the top and bottom removed.
- The resulting volume of the cylindrical shell is the surface area
of the cylinder times the thickness of the cylinder
- The formula for the volume will be: V =
,
where 2πx · f(x) is the surface area of the cylinder shell and
dx is its thickness
* Lets solve the problem
∵ y = 
∵ The plane region is revolving about the y-axis
∵ y = 32 and x = 0
- Lets find the volume by the shell method
- The definite integral are x = 0 and the value of x when y = 32
- Lets find the value of x when y = 0
∵ 
∵ y = 32
∴ 
- We will use this rule to find x, if
, where c
is a constant
∴ 
∴ The definite integral are x = 0 , x = 4
- Now we will use the rule
∵ 
∵ y = f(x) = x^(5/2) , a = 4 , b = 0
∴ 
- simplify x(x^5/2) by adding their power
∴ 
- The rule of integration of 
∴
from x = 0 to x = 4
∴
from x = 0 to x = 4
- Substitute x = 4 and x = 0
∴ ![V=2\pi[\frac{2}{9}(4)^{\frac{9}{2}}-\frac{2}{9}(0)^{\frac{9}{2}}}]=2\pi[\frac{1024}{9}-0]](https://tex.z-dn.net/?f=V%3D2%5Cpi%5B%5Cfrac%7B2%7D%7B9%7D%284%29%5E%7B%5Cfrac%7B9%7D%7B2%7D%7D-%5Cfrac%7B2%7D%7B9%7D%280%29%5E%7B%5Cfrac%7B9%7D%7B2%7D%7D%7D%5D%3D2%5Cpi%5B%5Cfrac%7B1024%7D%7B9%7D-0%5D)
∴ 
* The volume of the solid is 714.887 units³