Question

Anybody know how to do surface integrals?

Determine the area of the part of the paraboloid f(x; y) = a2 - x2 - y2 (a is a
positive constant) above the xy-plane. Hint: Use polar coordinates.

I'm not sure where to start

Answer 1

\[\int\limits _{S}\int\limits f(x,y,z) ds = \int\limits _{D}\int\limits f(x,y, g(x,y))\sqrt{(\frac{ \delta z }{ \delta x })^{2}+(\frac{ \delta z }{ \delta y })^{2}+1}dA\]

Answer 2

this is the surface integral in 3d, and you want it in an x,y plane

Answer 3

you'll find parametric equations, and the hint to use polar form to transform from the xy world to polar world, in order to simplify the double integral that you are dealing with

Answer 4

after you've made those simplifications, then start your surface integral using the new parametric equation set in the polar form

Answer 5

so now your S is part of the polar system

Answer 6

this is what the hint means,
\[x = r \cos (\theta) \\
y = r \sin(\theta)\]

Answer 7

you may also use the jacobian to help simplify calculation, but you may use whatever method you are used to, in order to solve the problem

Answer 8

i will be away from the computer a bit, when i come back i will help further.

Answer 9

me too, I have to make dinner, I'm still little confused, I'll try and work it a bit more.

Answer 10

don't worry you can finish your work, i will leave the solution for you by today or tomorrow, because is too late here.

Answer 11

thanks!!

Answer 12

you are welcome

Answer 13

After trying to work the problem I am still confused on where to start. These surface integrals are really confusing me for some reason. I'm not sure which function we are trying to parametrize or why.

I understand how to do the integration, just not how to set it up.

So in this problem we are trying to find the area of the paraboloid above the xy plane. The rest of this thing is a mystery.