Question

calc help

Answer 1

@jim_thompson5910

Answer 2

what does the graph look like for this one

Answer 3

we're dealing with this green region

Answer 4

the limits are from 1 to e^2

Answer 5

we revolve the green region around the line y = -2 to generate the 3D solid of revolution

Answer 6

and use the shell method?

Answer 7

I'd use the disk/washer method

Answer 8

so use washer?

Answer 9

yeah think of a bunch of washers stacked to form this 3D solid

Answer 10

kind of like this

http://www.mathdemos.org/mathdemos/washermethod/gallery/washers_8_allshells_static.gif

Answer 11

so the formula for the washer method is pi integral a to b [R(y)]^2-[r(y)]^2dy

Answer 12

wait but since it revolves around a line other then the axis wouldn't we use the formula pi integral a to b (outer radius)^2-(inner radius)^2?

Answer 13

what is the distance from y = -2 to y = 2?

Answer 14

4

Answer 15

4 is the outer radius

Answer 16

what is the distance from y = -2 to the curve y = ln(x)

hint: this is not a fixed number (it's in terms of x)

Answer 17

ln(x)+2?

Answer 18

yep ln(x) - (-2) = ln(x)+2

Answer 19

outer radius = 4
inner radius = ln(x)+2

Answer 20

so just square each of those? and subtract?

Answer 21

R = outer radius = 4
r = inner radius = ln(x)+2
V = volume of solid
a = 1
b = e^2

\[\Large V = \pi\int_{a}^{b}\left[\left(\text{outer radius}\right)^2-\left(\text{inner radius}\right)^2\right]dx\]
\[\Large V = \pi\int_{a}^{b}\left[R^2-r^2\right]dx\]
\[\Large V = \pi\int_{1}^{e^2}\left[4^2-\left(\ln(x)+2\right)^2\right]dx\]
\[\Large V = \pi\int_{1}^{e^2}\left[16-\left(\left[\ln(x)\right]^2+4\ln(x)+4\right)\right]dx\]
\[\Large V = \pi\int_{1}^{e^2}\left[16-\left[\ln(x)\right]^2-4\ln(x)-4\right]dx\]
I'll let you finish

Answer 22

i got 148.56

Answer 23

times pi

Answer 24

I'm getting a smaller value

Answer 25

now im getting 97.446 times pi

Answer 26

still too large

Answer 27

I'm getting approximately \(\Large 30.33433659\pi\)

Answer 28

http://www.wolframalpha.com/input/?i=int(4%5E2-(log(x)%2B2)%5E2,+x+%3D+1+..+exp(2))

Answer 29

i did it by hand and now am getting 12pi

Answer 30

how did you get 12?

Answer 31

wait nvmd i forgot to take the antiderivative

Answer 32

if you're curious, the exact answer is \(\Large \pi(6e^2-14)\)

Answer 33

yeah i got approx 95.297 now. i used the calculator. and that's around what you got after dividing pi

Answer 34

thanks!

Answer 35

no problem