Surface integral without using Gauss's Theorem

[mattmatt321]

Homework Statement

Find the value of the surface integral $$\int$$A $$\bullet$$ da, where A = xi - yj + zk, over the surface defined by the cylinder c² = x² + y². The height of the cylinder is h.

Homework Equations

I found the answer quite easily using Gauss's theorem, as the divergence of the vector A is simply 1, so the volume integral reduces to $$\int$$dv, which just becomes the volume of the cylinder. However, I was wondering how to integrate directly without using Gauss's theorem; i.e., integrate the original surface integral $$\int$$A $$\bullet$$ da. I feel like this is a pretty simple question and I'm thinking way too hard.

 

[gabbagabbahey]

I found the answer quite easily using Gauss's theorem, as the divergence of the vector A is simply 1, so the volume integral reduces to $$\int$$dv, which just becomes the volume of the cylinder. However, I was wondering how to integrate directly without using Gauss's theorem; i.e., integrate the original surface integral $$\int$$A $$\bullet$$ da. I feel like this is a pretty simple question and I'm thinking way too hard.

Divide the cylinder's surface up into 3 pieces: two endcaps and one curved surface. What is $d\textbf{a}$ for each of these 3 pieces? What variables change over each surface, which stay the same (and what are their fixed values)?