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!!!

- anonymous

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- amistre64

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

- amistre64

y = 4 -2x -x^2
y = x+4 and spin that around the x axis (y=0)

- amistre64

is your interval good?
6-2x-x^2 -x-6
-x^2 -3x = 0
-x(x+3) = 0; x= [-3,0] is good

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## More answers

- anonymous

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

- amistre64

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]

- amistre64

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

- amistre64

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

- amistre64

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

- anonymous

That was my radius

- amistre64

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

- amistre64

yours is -2 larger

- amistre64

you see it?

- amistre64

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

- amistre64

4 -2x -x^2 -[x+4]
-3x -x^2 right?

- anonymous

so I need to subtact 2 from both functions?

- amistre64

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]

- anonymous

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

- amistre64

you gotta squre your radius

- amistre64

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

- amistre64

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

- anonymous

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

- anonymous

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

- amistre64

-3x(-3x) = 9x^2
-x^2(-x^2) = x^4
-3x(-x^2)(2) = +6x^3

- amistre64

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

- amistre64

but then we get - everything

- anonymous

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

- amistre64

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

- amistre64

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

- anonymous

yes

- anonymous

I get 8.1

- anonymous

8.1 pi

- amistre64

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

- amistre64

which is 8.1 pi :)

- anonymous

So do you think that's right?

- amistre64

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

- anonymous

Thanks so much!!!

- amistre64

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

- amistre64

youre welcome :)

- anonymous

Nice way to check! Thanks again!

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