Question

surface integral

Answer 1

http://gyazo.com/91aa100d0362bbf5a1ec7c1246679645

Answer 2

\[\large \hat{n} dS = \langle -f_x, -f_y, 1\rangle \]

Answer 3

\(z = f(x,y) = 4-y^2\)

Answer 4

i dont know the above formula..

Answer 5

im trying to parametrise it

Answer 6

we need the normal vector to take dot product right ?

Answer 7

Correct formula :
\[\large \hat{n} dS = \langle -f_x, -f_y, 1\rangle dx dy\]

Answer 8

\(z = f(x,y) = 4-y^2\)

\(f_x = 0\)
\(f_y = -2y\)

\(\large \implies \ \hat{n} dS = \langle 0, 2y, 1\rangle dx dy\)

Answer 9

\[\large \iint_S \overrightarrow{F} \bullet \hat{n} dS = \iint_S\langle x^2-x, -xy, 3z \rangle \bullet \langle 0,2y,1 \rangle dx dy \]

Answer 10

we can replace z by 4-y^2 <<<<< this is the parameterization :)

Answer 11

\[\large \iint_S \overrightarrow{F} \bullet \hat{n} dS = \iint_S\langle x^2-x, -xy, 3(\color{red}{4-y^2}) \rangle \bullet \langle 0,2y,1 \rangle dx dy \]

Answer 12

\[\int \int ||n||dudv\] isnt this the formula for surfaces?

sorry, im not good with this

Answer 13

aahh yes you will get the same normal vector, but that would be like deriving the entire formula for each and every problem... we can use the formula for ndS directly

Answer 14

would the formula work for every surface integral? cos i havent seen it in my lecture notes

Answer 15

it works for every surface integral, we can derive it if you want...

just notice the limitation of this formula : we need to isolate z

Answer 16

surface need to be in z = f(x, y) form for using this formula - thats the only requirement

Answer 17

alright thanks
back to the question, would the integral limits be -2 < y < 2 , 0 < x <3 ?

Answer 18

lets see, its just a double integral

Answer 19

\[\large \iint_S \overrightarrow{F} \bullet \hat{n} dS = \iint_S\langle x^2-x, -xy, 3(\color{red}{4-y^2}) \rangle \bullet \langle 0,2y,1 \rangle dx dy\]

\[\large = \iint_S -2xy^2 -3y^2 +12 ~ dx dy\]

Answer 20

we're left with this after taking the dot product right ?

Answer 21

right... ;)

Answer 22

yes bounds look right !