Question

how do i finds the limits for this??

Using the CYLINDRICAL SHELL METHOD, find the volume generated by rotating the region boiunded by \(xy = 1\), \(x = 0\), \(y = 1\) and \(y = 3\) about the x-axis.

Answer 1

pls check the equation i used as well

\[2\pi \int_1^3 (y+1)(\frac{1}{y}) dy\]

@SmoothMath

Answer 2

Drawing mostly for my own visualization...

Answer 3

yeah we got the same drawings :p

Answer 4

Yes. I agree with that integral.

Answer 5

so the limits are the boundaries huh...but why didnt i get the same answer in my book =_=

Answer 6

\[2\pi [\int_1^3 dy + \int_1^3\frac{dy}{y}]\] that right?

Answer 7

Ooooh haha they are equivalent, but it's hard to see it.

Answer 8

\[\int\limits(y+1)(\frac{1}{y})dy = \int\limits(\frac{y}{y}) + \frac{1}{y}dy = \int\limits(1 + \frac{1}{y})dy\]

Answer 9

\[= \int\limits1dy + \int\limits(\frac{1}{y}) dy = \int\limits dy + \int\limits \frac{dy}{y}\]

Answer 10

then if i take the integral and evaluate the limits..it gives me \(4\pi + 2\pi(\ln 3)\) but the answer is supposedly \(4\pi\) only

Answer 11

Ooooh. We made a mistake. Take another look at the radius of the cylinder.

Answer 12

it's not y+1?? o.O

Answer 13

haha nope.

Answer 14

then what??

Answer 15

and how do you find it =_=

Answer 16

y

Answer 17

Rotated about the x-axis. So the radius is just the difference from the x axis... which is just y. Not y+1.

Answer 18

this hurts me so much :(

Answer 19

what do you mean difference from the x axis