determining whether to isolate x or y and formula for finding volume

[lime stirrup]

can someone help me determine when to get a function in terms of x or y? i also want to know when to use which formula for finding the volume.

[kindred sierra]

Hello!

It really depends around what axis you are rotating the function about. If you are rotating a function f about the x-axis, then you want it in terms of x. Similarly, for the y-axis you want it in terms of y.

[lime stirrup]

hi! thanks much for replying but when I checked the solution this is the way my professor did it. im sry if i take more of ur time but would you just keep in mind to change bounds when doing it in terms of the opposite axis?

the formulas and steps seem so straight forward but i keep getting mixed up when choosing the radius and height as well and applying it to the formulas

[kindred sierra]

The bounds change depending on what the function’s independent variable is, yes the bounds must change

[lime stirrup]

i see

tyty i will keep that in mind

[kindred sierra]

Did you have another question?

[lime stirrup]

yea.. if it wont bother you...

would you know when to use each formula? and if so, is the best way to just visualize the function?

this is what she showed; however even visualizing, its confusing to decide which formula to use

like this 1st picture should have a height as well no? sorry.. im pretty lost on this topic

genuinely sorryy for asking for so much my exam is today and i'm still so lost 😅

[kindred sierra]

Sure, so generally speaking, the disk and washer technique is a very general way of solving these types of problems. If you are revolving about an axis and can express your equation in terms of the same variable of the axis you are revolving about, then the disk/washer technique is viable

If this cannot be done, use the other one

This means if you are revolving about the x axis, you should express your integrand in terms of x

[lime stirrup]

i think i see now..

[kindred sierra]

We can do one

For 2

$\int_{0}^{2} (y^2 + 4) , dy$

[drifting junco]

Klaradiant

[kindred sierra]

This is the setup for it

I forgot a pi but who cares

And a square

But that’s the integrand

You would put that function for your integrand because it’s in terms of y, and you revolve about the y-axis. Disk method here

[lime stirrup]

oh

would u say my teacher did it in terms of x just to show another method?

[lime stirrup]

^ or am i totally wrong 😆

[kindred sierra]

Yes because your professor used the shell method it appears. For that, you would have to express it in terms of x instead

[lime stirrup]

i see

[kindred sierra]

Honestly, the shell method is for the birds

[lime stirrup]

okay i was mostly confused on the method she used for that question

loll

i do understand more now though

[kindred sierra]

The only time it’s useful is when you rotate about an axis that isn’t the x axis or y axis. It’s like a line like x = 15

And honestly, when you get to more advanced maths you can just do this with double integrals and triple integrals instead and be happy

Nobody cares enough about this stuff lol

[lime stirrup]

😅

lolol someday i will understand

ty again, id like to take no more of ur time

[kindred sierra]

You can forget this whole method later on but for now good luck on your exam

[lime stirrup]

🙏

i hope all goes well for you as well

.close

[void hare]

Solved

Post marked as solved by @whole mantle.

Use .unsolved if this was a mistake.

[lime stirrup]

@kindred sierra was this what i was missing xdd i never got that into my head until now

[kindred sierra]

That’s one part of it yes

[lime stirrup]

ok it actually made it so much easier

i appreciate it once again

disregarding finding which formula it never clicked that y = f(x) and x = f(y) literally

[kindred sierra]

I see

Something I should add.

If the shape is a cucumber then use disk. Anything else use shell.

[lime stirrup]

will keep that in mind.