00:01
Div f which is the divergence div f it's equal to zero plus two z plus eight z equal to ten z so using cylindrical coordinates cylindrical coordinates coordinates we have the triple integral integral e div f dv it's equal to triple integral integral here e same z dv and this is equal to the integral to the integral from 0 to 2 you have from 0 to 3 then from r squared to 9 10 z you have r d z d teta so you first integrate with respect to z you have the integral from 0 to 2 by 0 to 3 this will give us 405r minus 5 r to the power 5 the r d theta then let's integrate with respect to r and we have so we have 0 to 2 pi the theta then you have 0 to 3 you have 4 105 r minus 5 out squared the r so this gives us two pi then this will give us 3 ,645 divided by 2 minus 1 ,215 divided by 2 and this would be equal to 2 ,430 pi then we look at so on s 1 and s 1 the surface s 1 so the surface s 1 so s 1 the surface z it's equal to x squared plus y squared we have x squared plus y squared less than or equal to 9 with downward orientation then this is with downward orientation and f you have x y z to be equal to you have y squared z cube i plus two y z g plus four z squared key for that so then our so we take the integral over s 1 so you have f the s this is going to be minus then see go over the region d you have minus y squared z cube 2x minus 2y z you have times 2y x then plus 4 z squared so this will give us the integral over d.
04:36
So this thing is equal to 2 x, y squared.
04:43
You have x squared plus y squared to the power 3 plus 4 y squared.
04:53
You have x squared plus y squared minus 4.
04:59
This will give us x squared plus y squared to the power to d a so then in polar coordinates this then it's equal to so this then is going to be equal to the integral to the integral to the integral from 0 to 2 pi then from 0 to 3 you have 2 r cube cost theta sine squared theta times out to the power 6 plus 4 out squared sine square theta that's r squared minus four r to the power four we have r the r d theta so let's simplify what we have in the brackets this is zero to two by zero to so this is giving us 2 out to the power 10, sine square theta, cost theta, plus 4 out to the power 5, sine square theta.
06:50
So then, is then a thing is then a equal to 0 to 2 pi you first integrate with respect to r and that will give us 354 ,294 divided by 11 you have sign square theta plus 286 square theta minus 486 d theta then integrate this with respect to teta and this will give us minus 486 high so then this then implies that our surface so then the surface the surface the surface z is equal to nine so you have for the surface z equal to nine that is on s2 so let's say for s2 s2 s2 the surface z is equal to nine so you have x squared plus y squared less than or equal to 9 this is this was downward orientation so then this is an upward orientation so then if x y z is equal to you have y squared z plus 2 y squared zg plus 4 z squared k so then this gives us the integral over s 2 f the s is going to be so we have a double integral over d this gives us minus y squared you have z cube times zero minus 2 y z times zero plus 4 z squared d a d so then you have the double integral over d so this goes to zero this goes to zero as well z it's equal to nine so you have four times nine squared squared d .a...