# Area of a surface inside a cylinder

• Nov 26th 2010, 02:50 PM
jegues
Area of a surface inside a cylinder
Find the area of that part of the surface z = xy inside the cylinder x^2 +y^2 = a^2.

Well I don't know what the surface looks like but I have an idea of what the cylinder looks like.

It's like drawing a circle in the xy plane centered at the origin with radius a and pulling that circle in the z direction to create a cylinder.

Again, I don't know what surface looks like but the area of the surface should be

$\displaystyle \int\int_{S_{xy}} \sqrt{1 + (\frac{\partial z}{\partial x})^{2} + (\frac{\partial z}{\partial y}})^{2}}dA.$

If I take the partials accordingly and plug them in I get

$\displaystyle \int \int_{S_{xy}} \sqrt{1 + x^{2} + y^{2}}}dA$

now I'm stuck.

Any ideas?
• Nov 26th 2010, 02:52 PM
dwsmith
Spherical coordinates $\displaystyle \displaystyle x^2+y^2=r^2$. Then you can do trig sub $\displaystyle \displaystyle 1+tan^2=sec^2$
• Nov 26th 2010, 03:04 PM
jegues
Quote:

Originally Posted by dwsmith
Spherical coordinates $\displaystyle \displaystyle x^2+y^2=r^2$. Then you can do trig sub $\displaystyle \displaystyle 1+tan^2=sec^2$

We haven't covered Spherical coordinates in our course yet, couldn't I obtain the some results with polar coordinates?
• Nov 26th 2010, 03:05 PM
dwsmith
How have you not covered spherical if you have done polar?