This is an example I did in class for Section 13.6 on Surface Area.

Let z=f(x,y)=3x+y². Find surface area over the triangular region with vertices (0,0), (0,1), (1,1) in the xy-plane.

First, we plot the surface.

>	with(plottools):with(plots):

>	graph:=plot3d(3*x+y^2,x=0..y,y=0..1):

>	l1:=line([0,0,0],[0,1,0],color=blue,thickness=3): l2:=line([0,0,0],[1,1,0],color=blue,thickness=3): l3:=line([1,1,0],[0,1,0],color=blue,thickness=3): l4:=line([1,1,0],[1,1,4],color=black,linestyle=3,thickness=3): l5:=line([0,1,0],[0,1,1],color=black,linestyle=3,thickness=3):

>	display(graph,l1,l2,l3,l4,l5);

To compute the surface area, we find f_x = 3 and f_y = 2y. Thus the integrand is sqrt(3² + (2y)² + 1).

>	int(int(sqrt(4*y^2+10),x=0..y),y=0..1);

>

evalf(%);