Answer:
The volume is
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:
where 2πx · f(x) is the surface area of the cylinder shell and dx is its
thickness
* Lets solve the problem
- To find the volume V generated by rotating the region bounded
by the curves y = 4e^x and y = 4e^-x about the y-axis by use
cylindrical shells
- Consider that the height of the cylinder is y = (4e^x - 4e^-x)
- Consider that the radius of the cylinder is x
- The limits are x = 0 and x = 1
- Lets take 2π and 4 as a common factor out the integration
∴
∴
- To integrate and
we will use
integration by parts methods
∵ u = x
∴ u' = du/dx = 1 ⇒ differentiation x with respect to x is 1
∵ v' = dv/dx = e^x
- The integration e^x is e^x ÷ differentiation of x (1)
∴
∴
- Similar we will integrate xe^-x
∵ u = x
∴ u' = du/dx = 1
∵ v' = dv/dx = e^-x
- The integration e^-x is e^x ÷ differentiation of -x (-1)
∴
∴
∴ V = from 0 to 1
- Lets substitute x = 1 minus x = 0
∴
∴
∵
∴