Question

Let R be the region of the plane bounded by the curves y = 2^x and 6x − y = 8. The region R is rotated around the y-axis. Use the washer method to find an integral
whose value is the volume of the solid that you get.

Answer 1

where are you stuck?

Answer 2

from the start :(

Answer 3

can you find where the two curves intersect?

Answer 4

no i need help

Answer 5

y = 2^x and 6x − y = 8

6x − 2^x = 8

solve for x

Answer 6

x = (y+8)/6 ?

Answer 7

no

Answer 8

help i don't know?

Answer 9

you need to solve \[ 6x − 2^x = 8\] for x

Answer 10

you should get 2 numbers

Answer 11

I don't know how

Answer 12

you might be able to do it by inspection

Answer 13

what do i do

Answer 14

you either use a calculator or a computer...or just look at the equation and notice that the solutions are 'nice' numbers

Answer 15

so where do i go from 6x-2^x = 8

Answer 16

you can't combine them its 2 separate equations........y = 2^x and 6x-y=8

Answer 17

there is no algebraic way of solving this type of problem...use my privious post

Answer 18

idk how to even use your previous step

Answer 19

yes you can...to find where they cross each other

Answer 20

shouldn't it be x = log base 2 (y)
x = (y+8)/6

Answer 21

other than that idk what to do

Answer 22

I find it easier to see the solution for x first ...then it is easy to get the values of y

Answer 23

so can you do it for me since i have no idea what im doing?

Answer 24

notice that x=2 and x=4 satisfy the above equation

Answer 25

ok where

Answer 26

oh yes

Answer 27

plugging into \(y=2^x\) gives us that

y=4 and 16

Answer 28

so your integral will go from 4 to 16

Answer 29

x=4 you mean?

Answer 30

\(2^2=4\)
\(2^4=16\)

Answer 31

yes

Answer 32

\[2\le x\le 4\Rightarrow 4\le y\le 16\]

Answer 33

now just set up your integral

Answer 34

i need your help with that

Answer 35

you will have something of the form


\[\pi\int\limits_{a}^{b}\left(f(y)^2-g(y)^2\right)dy\]

where \(f(y)\) is the 'outer' function and \(g(y)\) in the inner function

Answer 36

why does the integral go from 4 to 16?

Answer 37

and isnt it 2pi

Answer 38

that would be for the method of cylindrical shells

Answer 39

you said to use the 'washer method'

Answer 40

so whats it gonna look like can you show me

Answer 41

you tell me

Answer 42

with everything plugged in

Answer 43

I've done all the work here...I'm done...your move

Answer 44

But i dont know what it is?

Answer 45

then you might want to look at some examples in your book

Answer 46

I already tried that why do you think i came here ?

Answer 47

I'm here to help...not to teach the entire topic...I teach during the day...not at night

Answer 48

so F(y) is 16?

Answer 49

no

Answer 50

i have no clue