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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
∴
∴
* The volume of the solid is 714.887 units³