Surface area of a part of z=xy in a cylinder?

[khfrekek92]

Homework Statement

What is the surface area of the part of z=xy that lies in the cylinder given by x^2+y^2=1?

Homework Equations

SA=(double integral) sqrt(Fx^2+Fy^2+1)dA

The Attempt at a Solution

I've graphed it but I don't know what to integrate or the bounds.. :/ any help? thank you!

 

[HallsofIvy]

You can write z= xy as the vector equation $\vec{r}(x,y)= x\vec{i}+ y\vec{j}+z\vec{k}= x\vec{i}+ y\vec{j}+ xy\vec{k}$. The two derivative vectors $\vec{r}_x= \vec{i}+ y\vec{k}$ and $\vec{r}_y= \vec{j}+ x\vec{k}$ are in the tangent plane at each point. Their cross product, $\vec{r}_x\times\vec{r}_y= -y\vec{i}- x\vec{j}+\vec{k}$, the "fundamental vector product" for the surface, is perpendicular to the surface and its length gives the "differential of surface area"- $dS= \sqrt{y^2+ x^2+ 1}dydx$.

Because the problem asks for the surface area of the part inside the cylinder $x^2+ y<br /> ^2= 1$, that circle is the boundary. You might want to put it in polar coordinates.

 

[khfrekek92]

Awesome I finally got it! Thanks so much for your help! :)