Finding Rate of Flow Outward Through a Paraboloid

[blacksoil]

1. Problem:
A fluid has density 1000 km/m3 and flows with velocity V =<x,y,z>, where x, y, and z are measured in meters, and the components of V are measured in meters per second. Find the rate of flow outward through the part of the paraboloid z = 64 - x^2 - y^2 that lies above the xy plane.


2. Relevant Equation:
Double Integral (Density*V ds)

Where V is the velocity function




3. The Attempt at a Solution
The only formula i found to solve this problem is formula above(2) which requires me to have the parametric equation for the paraboloid.. The thing is I really have no idea how to turn the paraboloid equation above into parametric equation

Helps are really appreciated...

 

[Raskolnikov]

I don't quite get understand the way you worded the problem, but here's the parametrized surface:

$$\vec{r}(x, \theta) = (xcos\theta, xsin\theta, 64 - x^2)$$

How is the fluid density measured in km/m^3? What units is your "rate of flow" supposed to be measured in? Also, are you sure your formula is correct?

 

[blacksoil]

i just copy pasted the actual problem from wamap.. and about the formula, I'm not sure either since i found it from google...
anyhow, how do you get the parametric equation? is that something to remember?

thanks

 

[Raskolnikov]

It's not something you have to memorize, but it is a method with which you should familiarize yourself. I'll give you a hint:

Fixing z traces out a circle on the paraboloid. The standard and obvious parametrization of the unit circle is $$\vec{r}(\theta) = (cos\theta, sin\theta).$$ Can you figure out what leads us to the final answer from here?