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 = [tex]\int\limits^a_b {2\pi xf(x)} \, dx[/tex],
where 2πx · f(x) is the surface area of the cylinder shell and
dx is its thickness
* Lets solve the problem
∵ y = [tex]x^{\frac{5}{2}}[/tex]
∵ 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
∵ [tex]y = x^{\frac{5}{2}}[/tex]
∵ y = 32
∴ [tex]32=x^{\frac{5}{2}}[/tex]
- We will use this rule to find x, if [tex]x^{\frac{a}{b}}=c, then=== x=c^{\frac{b}{a}}[/tex] , where c
is a constant
∴ [tex]x=(32)^{\frac{2}{5}}=4[/tex]
∴ The definite integral are x = 0 , x = 4
- Now we will use the rule
∵ [tex]V = \int\limits^a_b {2\pi}xf(x) \, dx[/tex]
∵ y = f(x) = x^(5/2) , a = 4 , b = 0
∴ [tex]V=2\pi \int\limits^4_0 {x}.x^{\frac{5}{2}}\, dx[/tex]
- simplify x(x^5/2) by adding their power
∴ [tex]V = 2\pi \int\limits^4_0 {x^{\frac{7}{2}}} \, dx[/tex]
- The rule of integration of [tex]x^{n} is ==== \frac{x^{n+1}}{(n+1)}[/tex]
∴ [tex]V = 2\pi \int\limits^4_0 {x^{\frac{9}{2}}} \, dx=2\pi[\frac{x^{\frac{9}{2}}}{\frac{9}{2}}][/tex] from x = 0 to x = 4
∴ [tex]V=2\pi[\frac{2}{9}x^{\frac{9}{2}}][/tex] from x = 0 to x = 4
- Substitute x = 4 and x = 0
∴ [tex]V=2\pi[\frac{2}{9}(4)^{\frac{9}{2}}-\frac{2}{9}(0)^{\frac{9}{2}}}]=2\pi[\frac{1024}{9}-0][/tex]
∴ [tex]V=\frac{2048}{9}\pi=714.887[/tex]
* The volume of the solid is 714.887 units³
