# What is the Volume of Revolution if the area bounded by the curve #y=x^2-4x# and the #x#-axis is is rotated about the #x#-axis?

##### 1 Answer

Mar 5, 2017

# (512 pi) /15 \ unit^3#

#### Explanation:

I recommend that you always draw a sketch to clarify what needs calculating.

graph{x^2-4x [-10, 10, -5, 5]}

The curve intersects the

# x^2-4x = 0 => x(x-4)=0 => x=0,4#

The Volume of Revolution about

# VOR = int_alpha^beta \ piy^2 \ dx #

So in this case:

# VOR = int_0^4 \ pi \ (x^2-4x)^2 \ dx #

# " " = pi \ int_0^4 \ (x^4-8x^3+16x^2) \ dx #

# " " = pi [ x^5/5-2x^4+16x^3/3]_0^4 #

# " " = pi { (4^5/5-2*4^4+16*4^3/3)-(0) } #

# " " = pi (1024/5-512+1024/3)-(0) } #

# " " = (512 pi) /15 #