Finding Volume of Solid of Revolution: y = (x−4)3,y=,x = 0,x = 5; about y-axis

[theletterm]

Hello everyone, I've been slamming my head on the desk trying to figure out this problem for a while now, and can't figure out where its going wrong. Any help would be appreciated!

"Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of the given equations about the indicated axis.
y = (x−4)³, the x-axis,x = 0,x = 5; about the y-axis"

I know the general theory and steps that entail solving this sort of problem, and what the graph for this looks like.

Started by putting (x-4)^3 into terms of y- getting y^(1/3)+4. Then I found the limits of integration- lower limit being -64 and upper being 1, and split the problem into the integral from -64 to 0, and 0 to 1.
Then both sides of the problem are squared.
(0 to -64)pi(y^2/3 + 4y^(1/3) +16) + (1 to 0)pi(5^2-(4+y^1/3)^2)
(^In this part I know that on the upper section from 1 to 0 that it needs to be in the washer method, and that could be where some of my problem is stemming from.)

Then I integrated.
pi(0 to -64) (3y^(5/3)/5 + 3y^(4/3) +16y + pi(1 to 0) (-3y^(5/3)/5 -3y^(4/3)+9y)

After that I input the limits of integration to get 4352pi/5+27pi/5 = 4379pi/4

I've reworked this problem about three times to check for mistakes and keep getting the above answer, so there's obviously something wrong with how I'm approaching this, given the homework website I'm using didn't accept it. I'm willing to post a photo of my work by hand if any of the notation that I used is hard to read.
Any help is appreciated, or a step by step breakdown of the solution.
Thanks so much!

 

[Harry_the_cat]

Hello everyone, I've been slamming my head on the desk trying to figure out this problem for a while now, and can't figure out where its going wrong. Any help would be appreciated!

"Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of the given equations about the indicated axis.
y = (x−4)³, the x-axis,x = 0,x = 5; about the y-axis"

I know the general theory and steps that entail solving this sort of problem, and what the graph for this looks like.

Started by putting (x-4)^3 into terms of y- getting y^(1/3)+4. Then I found the limits of integration- lower limit being -64 and upper being 1, and split the problem into the integral from -64 to 0, and 0 to 1.
Then both sides of the problem are squared.
(0 to -64)pi(y^2/3 + 4y^(1/3) +16) + (1 to 0)pi(5^2-(4+y^1/3)^2)
(^In this part I know that on the upper section from 1 to 0 that it needs to be in the washer method, and that could be where some of my problem is stemming from.)

Then I integrated.
pi(0 to -64) (3y^(5/3)/5 + 3y^(4/3) +16y + pi(1 to 0) (-3y^(5/3)/5 -3y^(4/3)+9y)

After that I input the limits of integration to get 4352pi/5+27pi/5 = 4379pi/4

I've reworked this problem about three times to check for mistakes and keep getting the above answer, so there's obviously something wrong with how I'm approaching this, given the homework website I'm using didn't accept it. I'm willing to post a photo of my work by hand if any of the notation that I used is hard to read.
Any help is appreciated, or a step by step breakdown of the solution.
Thanks so much!

I haven't worked the problem myself but you have made a mistake here:

Then both sides of the problem are squared.
(0 to -64)pi(y^2/3 + 4y^(1/3) +16) + (1 to 0)pi(5^2-(4+y^1/3)^2)

Shouldn't the first 4 be an 8??