finding volume of revolution

Question

I'm trying to find the volume of the revolution of equation 

 

y=0.0025x^3 {110<y<0}

 

around the y-axis, 

 

please can someone help???

Answer 1

Volume of solid revolution is found by

V= π∫[f(x)]²dx, (if rotation is around x -axis)

V= π∫[x]2dy, (if rotation is around y -axis)

Since in your question rotation is around y-axis, you have to express y in terms of x:

y=0.0025x³

x=(400y)^(1/3) 

The volume is calculated by

V=π∫x²dy=π∫(400y)^2/3dy. (*)

But not sure what integration  limits (upper and lower bounds).

You have 110<y<0, which is wrong. Once you have the correct bounds, you just calculate integral.

 

π∫(400y)^2/3dy=π400^2/3∫y^2/3dy=π400^2/3 (3/5) y^(5/3).