Question

Consider the solid obtained by rotating the region bounded by the given curves about the line y = 5.
y = 5x, y = sqrtx
find the volume

Answer 1

hmmm....we just did one like like this earlier :)

Answer 2

hello :)) ya its the exact same one. im a little stuck

Answer 3

y = 5; means that we take the functions and -5 to get them to y=0

Answer 4

5x-5 and sqrt(x)-5 rotated about the x axis

Answer 5

yeah, both of those are squared right. then i take the anti derivative of it correct ?

Answer 6

5x = sqrt(x)

25x^2 = x

25x^2 -x = 0

x(25x - 1) = 0

x = 0 and x = 1/25 right?

Answer 7

yup

Answer 8

then our bounds of integration are 0 to 1/25

Answer 9

im a little confused now because i thought our bounds were from 0 to 5 from before

Answer 10

before you had 5sqrt(x). Did you typo it here?

Answer 11

no thats correct but when we first graphed the two equations, werent the bounds from 0 to 5 ?

Answer 12

cant recall, been doing alot of them lately :)

but the steps are the same... i these equations are good; then the results will be

x = 0 and x = 1/25

Answer 13

then i do the same thing for 5x - 5
5x = 5
x = 1
so the bounds are from 0 to 1 for that part of the integral?

Answer 14

when does 5x = sqrt(x)

wehn x = 0 and x = 1/25

5/25 = 1/5 and sqrt(1/25) = 1/5 right?

Answer 15

yup

Answer 16

we arent double integrating this are we?

Answer 17

ohh nooo we arent. sorry sorry. i was mixing it up with something else

Answer 18

now we want to add up all the standard areas thru integration; and the area of each circle we generate by spinning is:

pi [f(x)-5]^2 right?

Answer 19

yes

Answer 20

we can do each function seperately and subtract them in the end; or do them together.

pi [f(x)-5]^2 - pi[g(x)-5]^2 will give us the total volume then

Answer 21

pi [S] [5x-5]^2 - [sqrt(x)-5]^2 dx ; [0,1/25]

Answer 22

25x^2 -50x +25 - [x-10sqrt(x) +25]

25x^2 -50x +25 - x+10sqrt(x) -25

pi [S] [25x^2 -51x +10sqrt(x)] dx ; [0,1/25] then right?

Answer 23

25x^3 51x^2 20sqrt(x^3)
------ - ------- + ----------
3 2 3

Answer 24

and then we plug in the boundaries, and subtract them right?

Answer 25

yes; but the easy thing here is that 0 = 0 so we only determine 1/25 for our answer

Answer 26

ah. gotcha. thank you so much for your help. made it a lot clearer for me. thank you :)

Answer 27

youre welcome :)

Answer 28

dot worry if the volume is negative; just take the absolute value of it then :)