Volumes by revolution NEED HELP!

[iwils0a]

Volumes by revolution NEED HELP PLEASE!

How do you find the volume when the graphs sort of "bound" two locations? I just need an explanation or starting point I think but I'm working with the functions x=y⁴/4 - y²/2 and x=y²/2 revolved about the line y=5

 

[iwils0a]

any assistance is appreciated! The graphs intersect at -2, 0 and -2 in terms of y

 

[HallsofIvy]

How do you find the volume when the graphs sort of "bound" two locations? I just need an explanation or starting point I think but I'm working with the functions x=y⁴/4 - y²/2 and x=y²/2 revolved about the line y=5

The first thing you need to do is find where those two graphs cross. One obvious way to do that is set "x= x", or, in this case, \(\displaystyle x= y^4/4- y^2/2= y^2/2\) so that we immediately have \(\displaystyle y^4/4- 2y^2/2= y^2(y^2/4- 1)= 0\). Or notice immediately that [itex]x= y^2/2[/itex] is the second term in the first equation and that \(\displaystyle x^2= (y^2/2)^2= y^2/4\) is the first term. That is, we can write \(\displaystyle x= x^2- x\) so that \(\displaystyle x^2- 2x=0\).

In the first case we get y= 0, y= 2, or y= -2. From that , since \(\displaystyle x= y^2/2\), x= 0 or x= 2. In the second, \(\displaystyle x^2- 2x= x(x- 2)\) we also get x= 0 and x= 2. It's easy to show that both \(\displaystyle x= y^4/4- y^2/2\) and \(\displaystyle x= y^2/2\) are both below y= 5, for x between -2 and 2 so this will involve a 'washer method' or equivalently, two "disk method" integrations, one for \(\displaystyle x= y^4/4- y^2/2\), the other for \(\displaystyle x= y^2/2\) and then subtracting.

 

[iwils0a]

Alright Thank you for that but I had already found the intersections what I meant by starting step and should have said was for solving the volume. both functions have two plot points for each x value so I'm not sure which function is considered the top or higher graph or also if I only do the positive or negative sides of the graph to fix the multiple plot points issue and then double it after I solve for the volume

 

[HallsofIvy]

Do both. It is the whole thing, both portions, that are being rotated. I recommend you do each separately and add.