Question

volume of solid of rotation

Answer 1

determine the volume of the solid obtained by rotating the region bounded by y=x^2-2x and x-axis, y=0, rotate about the y-axis

Answer 2

would it just be \[2 \pi \int\limits_{0}^{2}x^{2}-2xdx\]

Answer 3

@dtan5457

Answer 4

@freckles

Answer 5

I can tell you it isn't, it's close though. First explain how you arrived at that conclusion.

Answer 6

did I forget to multiply by x also?

Answer 7

You got the 2pi correct and the limits. It's the bit you're integrating. Do you recall the formula for volume of rotation?
\[2\pi \int\limits_{0}^{a}{y}^{2}dx\]

Answer 8

I was using cylindrical shell. Wouldn't that be disk method and be\[\pi \int\limits_{0}^{a}y^{2}dx\]

Answer 9

I understand the equation except the 2pi. I would think it would just be pi

Answer 10

Oh you know cylindrical shells? Can you post your working then?

Answer 11

the problem I have is that I have another problem it is the same question but it is about x=0 and I thought "cant I use the same method?" That would give me the sam answer though and my teacher said you would get different answers about different axes

Answer 12

here is my work though

Answer 13

\[V=\int\limits_{0}^{2}2 \pi x (x^{2}-2x)dx\]\[2 \pi \int\limits_{0}^{2}x^{3}-2x^{2}dx\]

Answer 14

\[2 \pi[\frac{ 1 }{ 4 }x^{4}-\frac{ 2 }{ 3 }x^{3}]\]

Answer 15

plugging in 2 I got \[\frac{ 8 \pi }{ 3 }\]

Answer 16

So, i understand you worked out for rotating against x=0, and used the same equation to generate your new answer?

Answer 17

yes but I from what I know when I rotate it about x=0 the answer should be different than about y=0

Answer 18

is it because I can only use cylindrical shells when rotating in relation to y axis?

Answer 19

Cylindrical shells works regardless of axes. Even oblique axes if you want to be really trippy.

I'm not quite understanding how you translated your answer for x = 0 to y = 0. I'll post a solution for cylindrical shells for y = 0 in a sec.

Answer 20

sorry i mean y axis

Answer 21

Wait i think there's an error

Answer 22

so how do I set it up if I want to do it about the x-axis rather than the y

Answer 23

So anyway I think my final answer is wrong, but that's how you set it up. Same kind of deal for x-axis. I'll draw you a diagram if you like.

Answer 24

i would appreciate that yes. Thank you

Answer 25

Hmm now that I come to drawing it I see you can only do it for y axis in this case.

Answer 26

Disk works though

Answer 27

ok how would I do disk for this?

Answer 28

Just a sec

Answer 29

Actually i think my answer was correct

Answer 30

So I think you're answer was indeed off by a factor of x. Sorry I haven't mathed in like 3 months :/

Answer 31

so you cant do cylindrical shells around the x for this but you can do disk?

Answer 32

now for my disk here is a copy of my work

Answer 33

Yeah that's it.

Answer 34

when can you do cylindrical shells for around x and how?

Answer 35

sorry for the sideways but it's number 1

Answer 36

is my work correct?

Answer 37

You can cylindrical for x axis, whenever your region doesn't touch the axis. It's pretty much just an extended version of disks.

Answer 38

All correct except last line. Should be 16/15 pi instead of 16/5 pi

Answer 39

Anyway hopefully this has been useful, sorry I have to go now.

Answer 40

oh ok thank you for your help

Answer 41

and it did help some I just need to ask my teacher a few questions now

Answer 42

Your welcome