Let $\mathcal{T}$ be the triangular region with vertices $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ oriented with upward-pointing normal vector.

A fluid flows with constant velocity field $\mathbf{v}=3 \mathbf{i}+8 \mathbf{j}~\mathrm{m/s}$. Calculate:
(a) The flow rate through $\mathcal{T}$
(b) The flow rate through the projection of $\mathcal{T}$ onto the $xy$-plane [the triangle with vertices $(0,0,0)$, $(1,0,0)$, and $(0,1,0)$]

Assume distances are in meters.

(a) $\iint_{\mathcal{T}} \mathbf{v}\cdot d\mathbf{S} =$
(b) $\iint_{\mathcal{T}} \mathbf{v}\cdot d\mathbf{S} =$