Question

Evaluate the surface integral s is the triangular region with vertices (1, 0, 0), (0, 5, 0), (0, 0, 5)

Answer 1

With no specific integrand given, I'm assuming you need only find the area of
`\mathcal S`, in which case we can parameterize the surface by


`\mathbf s(u,v)=(\langle1,0,0\rangle(1-u)+\langle0,5,0\rangle u)(1-v)+\langle0,0,5\rangle v`

`\mathbf s(u,v)=\langle(1-u)(1-v),5u(1-v),5v\rangle`

with
`0\le u\le1,0\le v\le1`. Then the area of
`\mathcal S` is given by


`\displaystyle\iint_(\mathcal S)\mathrm dS=\int_([0,1]^2)\|\mathbf s_u*\mathbf s_v\|\,\mathrm dv\,\mathrm du`

`\displaystyle=15\sqrt3\int_(u=0)^(u=1)\int_(v=0)^(v=1)(1-v)\,\mathrm dv\,\mathrm du=\frac{15\sqrt3}2`