Thread: volume of revolution

1. volume of revolution

how do you find the volume of the solid generated by revolving btw the x axis and y= 4x - x^2 about the line y=6.

the volume required is not bounded btw the line y= 6 and the eqn and hence i am stuck..
thanks!

2. Originally Posted by alexandrabel90
how do you find the volume of the solid generated by revolving btw the x axis and y= 4x - x^2 about the line y=6.

the volume required is not bounded btw the line y= 6 and the eqn and hence i am stuck..
thanks!
first find the points of intersection. then you can use the disk (washer) method to find the volume. do you remember the formula?

(by the way, graphing the region is almost always super helpful)

3. the point of intersection with the x-axis is x= 0 and x=4.

what i did was
find the volume of the cylinder of radius 4, height 6 - (pi) integrate (6-y)^2
= pi( 96) - pi integrate ( 6- 4x + x^2)^2

but i could not get the answer.

should it not be 6- y for the second part of my equation?

4. Originally Posted by alexandrabel90
the point of intersection with the x-axis is x= 0 and x=4.

what i did was
find the volume of the cylinder of radius 4, height 6 - (pi) integrate (6-y)^2
= pi( 96) - pi integrate ( 6- 4x + x^2)^2

but i could not get the answer.

should it not be 6- y for the second part of my equation?
actually, for the cylinder, the radius is 6 and the height 4. you want to integrate the outer radius squared minus the inner radius squared, and multiply all that by pi. the desired volume is given by:

$\displaystyle V = \pi \int_0^4 [6^2 - (6 - 4x + x^2)^2]~dx$