## anonymous 4 years ago The region bounded above by the line y=6, below by the curve y=sqrt(x) and to the left by the y-axis, rotated around x=36.

1. amistre64

we should get the same results if im thinking of it correctly

2. amistre64

but just in case; think of this as a solid cylinder that we are removing the under part of the sqrt(x)

3. amistre64

|dw:1327534530288:dw|

4. amistre64

so, cylindar itself; r=36, h=6; pi 36^2*6 is total area of this cylindar, now we gut out the area under the sqrt(x)

5. anonymous

so it's like an upside down bowl

6. amistre64

yes, and lets disc out the underside, i think it might integrate easier that way

7. anonymous

ok after I integrated I got, 18x^2 - 2/3 * x^(3/2)

8. amistre64

|dw:1327534763514:dw|

9. amistre64

r = 36-y^2; when y=0, r=36; when y=6, r=0 that works in my head lol

10. amistre64

V = int (pi [f(y)]^2) from 0 to 6 V = int (pi(36-y^2)^2) from 0 to 6, im just gonna paste this into wolfram to verify :)

11. anonymous

so I couldn't leave the y=sqrt(x), but would have to convert it to y^2 instead?

12. amistre64

yes, for disc; maybe shell is easier?

13. anonymous

Gotcha! So your r should always equal 0 if a limit is plugged in?

14. amistre64

not always, the radius changes as the integration moves across the interval; in this case, with the disc, the radius is determined by 36 over - y^2 back

15. amistre64

but lets do shell and see if my thoughts improve lol

16. anonymous

haha alright

17. amistre64

the shell moves it radius from 0 to 36 as it does, the height is determined by sqrt(x); but i tend to like to move everything to the yaxis to get a clearer picture; so; sqrt(x+36) moves everything to the left by 36 and gets our rotation around the y axis

18. amistre64

|dw:1327535297927:dw|

19. anonymous

Ahh. and that isn't what we want!

20. amistre64

this is another view of what we want; its better to me

21. amistre64

$V=2pi\int_{0}^{-36}x(\sqrt{x+36})dx$ i believe thats right;

22. amistre64

sorry, but I gotta get; hope i havent messed it up too bad. good luck tho :)

23. anonymous