A fluid with density 1230 flows with the velocity $\displaystyle V = y$i $\displaystyle +$ j $\displaystyle + z$k. Find the rate of flow (of mass) upward through the paraboloid $\displaystyle z = 9 - \frac{1}{4}(x^2 + y^2), x^2 + y^2 \leq 36$.
A fluid with density 1230 flows with the velocity $\displaystyle V = y$i $\displaystyle +$ j $\displaystyle + z$k. Find the rate of flow (of mass) upward through the paraboloid $\displaystyle z = 9 - \frac{1}{4}(x^2 + y^2), x^2 + y^2 \leq 36$.
the density is mass/volume
Flux is Volume/time
You want then mass/time = density*flux
N = x/2i + y/2 j + k
V*N = xy/2 + y/2 + z = xy/2 +y/2 + 9-1/4(x^2 +y^2)
convert to Polar coordinates
and integrate over the disk of radius 6
If you have trouble see the attachment in the following post