Use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the y-axis. y = x5/2 y = 32 x = 0

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

honesty mood

step-by-step explanation: