Surface Area: Examples

1. Find the surface area of the paraboloid z=1+x^2+y^2 that lise above the unit circle.

Anwser:

Because f_x=2*x and  f_y=2*y you would get the following integral for surface area:

>	Int(Int(sqrt(1+4*x^2+4*y^2),x),y);

To make this problem easier to solve we can convert into polar coordinates:

>	int(int(sqrt(1.0+4*r^2)*r,r=0..1),theta=0..2*Pi);

2. Find the surface area S of the portion of the hemisphere f=sqrt(25-x^2-y^2) that lise above the region R bounded by the circle x^2+y^2<9

Anwser:

First find the partial derivates of

>	diff(sqrt(25-x^2-y^2),x);

>	diff(sqrt(25-x^2-y^2),y);

We then set up the surface area integral and simplify:

>	sqrt(1+(-x/(25-x^2-y^2)^(1/2))^2+(-y/(25-x^2-y^2)^(1/2))^2);

>	simplify(%);

Next we convert to polar coordinates and solve:

>	int(int(5*(-1/(-25+r^2))^(1/2)*r,r=0..3),theta=0..2*Pi);