# Evaluating $\iint_s \vec F \cdot \hat n ds$ around the curved surface of cylinder cut by plane at $45^\circ$

I need to calculate the surface integral of $F(x,y,z) = \hat i x +\hat j y + \hat k z$ on the curved part of surface $x^2+z^2 = 1, x+y=2,$ and $y$ goes from $1$ to $3$ as shown in following figure. How do I evaluate $\displaystyle \iint_S \vec F .\hat n ds$ this surface?

EDIT::I couldn't do it via parametrization, I got the above figure which is incorrect. Using this formula $\iint_s \vec F \cdot \frac{\nabla \phi }{|\nabla \phi|}\sqrt{1 + (z_x)^2 + (z_y)^2} dx dy$ I got the following. Not sure if it's correct. $$\int_1^3 \;dy \int_{-1}^{2-y} \vec F(x, y , \sqrt{1-x^2})\cdot \frac{x \hat i + \sqrt{1-x^2}\hat k}{\sqrt{1-x^2}} dx \\ + \int_1^3 \;dy \int_{-1}^{2-y} \vec F(x, y , -\sqrt{1-x^2})\cdot \frac{x \hat i - \sqrt{1-x^2}\hat k}{\sqrt{1-x^2}} dx$$

-

I tried that, that's where I got that above graph. $(\cos (\theta), \sin (\theta), t)$ but I was having problem with bounds for $\theta$. Check the revision – hasExams May 31 '13 at 12:47