Question

Find the volume of a solid generated by revolving the graphs of the functions y=6-2x-x^2 and y=x+6 about the line y=2
Intersections points are x=-3 and x=0

I think the radius of the solid should be 6-2x-x^2-(x+6)-2 which is -2-3x-x^2

then I have Vol = pi*Integral from -3 to 0 of (-2-3x-x^2)^2dx
I integrate and get pi *[4x65/5+3/2x^4+13x^3/3+6x^2+4x ] evalulated between -3 and 0. I end up with 147.9pi which seems way too big.Can you find my mistake? Thank you so much!!!

Answer 1

if i see it right; you went ahead and subtracted 2 to get it to the x axis right?

Answer 2

y = 4 -2x -x^2

y = x+4 and spin that around the x axis (y=0)

Answer 3

is your interval good?

6-2x-x^2 -x-6

-x^2 -3x = 0

-x(x+3) = 0; x= [-3,0] is good

Answer 4

Since it is being revolved around y=2, I was subtracting 2 to eliminate that much verticasl distance from the radius.

Answer 5

then we can integrate:

subtract one function from the other...

4-2x-x^2 - x-4

pi [S] [-x^3 -3x]^2 dx ; [-3,0]

Answer 6

-x^2....typoed it :)

Answer 7

pi [S] (x^6 +9x^2 +6x^4) dx right?

Answer 8

-2-3x-x^2 is your function; why?

Answer 9

That was my radius

Answer 10

it doesnt match my radius tho :) so whos right?

Answer 11

yours is -2 larger

Answer 12

you see it?

Answer 13

y=6-2x-x^2 (-2) = 4-2x -x^2

y=x+6 (-2) = x+4

Answer 14

4 -2x -x^2 -[x+4]

-3x -x^2 right?

Answer 15

so I need to subtact 2 from both functions?

Answer 16

Yes, becasue you need to move both function

pi [S] (x^6 +9x^2 +6x^4) dx

pi (x^7/7 +3x^3 + 6x^5/5) ; [-3,0]

Answer 17

How are you getting thisx^6 +9x^2 +6x^4) dx

Answer 18

you gotta squre your radius

Answer 19

pi [S] [f(x)]^2 dx

Answer 20

[f(x)]^2 = [-2x-x^2]^2

Answer 21

If the radius is (-3x-x^2) and we square it, don't we get 9x^2-6x^3=x^4?

Answer 22

9x^2-6x^3+x^4

Answer 23

-3x(-3x) = 9x^2

-x^2(-x^2) = x^4

-3x(-x^2)(2) = +6x^3

Answer 24

we could go with -1(3x+x^2) as well i htink

Answer 25

but then we get - everything

Answer 26

That 's OK b/c the def integral goes form -3 to 0

Answer 27

wanna go with - everything? i dont think it makes a diff :)

Answer 28

pi (-3x^3 -x^5/5 -3x^4/2) then right?

Answer 29

yes

Answer 30

I get 8.1

Answer 31

8.1 pi

Answer 32

pi * (((-3) * ((-3)^3)) - (((-3)^5) / 5) - ((3 * ((-3)^4)) / 2)) = 25.4469005 is what google spits out ....

Answer 33

which is 8.1 pi :)

Answer 34

So do you think that's right?

Answer 35

i think so; you can double check it if you want.... but im pretty sure we worked the kinks out of it :)

Answer 36

Thanks so much!!!

Answer 37

try to integrate one function at a time and subtract one from the other in the end to verify; kinda like making a donut and then cutting out the hole

Answer 38

youre welcome :)

Answer 39

Nice way to check! Thanks again!