Help on script for parametric surface integral

I'm working on a short script that I can repeatedly use to solve parametric surface integrals. There is a post on this elsewhere on MSE, but it is a complicated doublecontourintegral function. It is impossible to parse for a newcomer like me, and it seems possible to create something more simple.

Here's an example problem:

Find the surface integral of $$(x + y + s)dS$$, where S is the parallelogram with parametric equations $$x = u + v$$; $$y = u - v$$; $$z = 1 + 2u + v$$; and $$0 \leq u \leq 2$$, $$0 \leq v \leq 1$$.

So I wanted to write a script that I can just type in the non-parametrized equation, i.e., $$(x + y + s)$$ in this example. Then I'd enter in the parameters for x, y, and z (given in this problem).

Next, I'd take the partial derivatives with respect to u an v of the parametrized equation.

Next, I'd cross product those two partial derivatives and find its magnitude, using Norm.

Next, I'd re-parametrize $$(x + y + s)$$.

Finally, I'd integrate the re-paremetrized surface, multiplied by the magnitude of the partials, over the region, given in this problem as $$0 \leq u \leq 2$$, $$0 \leq v \leq 1$$.

I am new to Mathematica, and I started this sequence with the following code, but it doesn't seem to work. I'd welcome any help.

Clear[x, y, u, v]
x[u_, v_] := u + v;
y[u_, v_ ] := u - v;
z[u_, v_] := 1 + 2 u + v;
R[u_, v_] := {x[u_, v_], y[u_, v_], z[u_, v_]}


UPDATE: This code works well on problems I've tried. But I'm having trouble with another problem. The book answer is 1/(10sqrt(2)). I'm attaching as a picture its worked example. But both of your methods give 1/5sqrt(2). Did I miss something? Is the book in error? Here's the code I used, and the book's answer.

x[u_, v_] := u Cos[v];
y[u_, v_] := u Sin[v];
z[u_, v_] := u;
r[u_, v_] := {x[u, v], y[u, v], z[u, v]};
reg = ParametricRegion[r[u, v], {{u, 0, 1}, {v, 0, Pi/2}}];
Integrate[x y z, {x, y, z} \[Element] reg]


And here's is the book answer, which is Problem 6:

• 1/(5 Sqrt[2]) == Sqrt[2]/10 – cvgmt Dec 28 '20 at 4:37
• Of course it is. Drrr. Apologies. – JDVC Dec 28 '20 at 4:39

x[u_, v_] := u + v;
y[u_, v_] := u - v;
z[u_, v_] := 1 + 2 u + v;
r[u_, v_] := {x[u, v], y[u, v], z[u, v]};
Integrate[(x[u, v] + y[u, v] + z[u, v]) Norm@
Cross[D[r[u, v], u], D[r[u, v], v]], {u, 0, 2}, {v, 0, 1}]


11 Sqrt[14]

Or

x[u_, v_] := u + v;
y[u_, v_] := u - v;
z[u_, v_] := 1 + 2 u + v;
r[u_, v_] := {x[u, v], y[u, v], z[u, v]};
reg =ParametricRegion[r[u, v], {{u, 0, 2}, {v, 0, 1}}];
Integrate[x + y + z, {x, y, z} ∈ reg]


11 Sqrt[14]

• Thanks so much for the code. It is very helpful. See my update above, if you can, and perhaps you can clarify? – JDVC Dec 28 '20 at 4:21
• And this code doesn't seem (?) to work if the parameters are trig functions. I get strange message when I use trig functions in the parameters. Is that right? – JDVC Dec 28 '20 at 4:34
• @BRC You can use Clear["`*"]; to clear the variable before the new evaluation. – cvgmt Dec 28 '20 at 4:39
• I tried that, alas. If you make the integrand something with just one variable, would that make the code unworkable? For example, in a similar problem to "Update' above, I changed the integrand to just "y" and it gave me gobbledygook: \!(*UnderscriptBox[([Integral]), ({x, y, z} [Element] ParametricRegion[{{Cos[v]\ u, u\ Sin[v], v}, 0 [LessEqual] u [LessEqual] 1 && 0 [LessEqual] v [LessEqual] *FractionBox[([Pi]), (2)]}, {u, v}])])y – JDVC Dec 28 '20 at 4:51
• Perhaps if I just have a single (or even two )variable in the integrand (all of the other problems had x, y, and z), I need some kind of placeholder for the missing variable? – JDVC Dec 28 '20 at 4:55