(865, #7) Find the surface area of the part of the surface z = x y that lies within the cylinder x² +y² = 1..

Solution: Depending on your point of view, the surface is either a saddle or a potato chip:

f := (x,y)-> x*y:
plot3d([r*cos(t), r*sin(t), r^2*cos(t)*sin(t)],
r = 0..1, t = 0..2*Pi, color = navy, grid = [6, 25]);

Its surface area is equal to (2/3) Pi (2^1/2 - 1):

g := (x,y) -> 1 + diff(f(x,y),x)^2 + diff(f(x,y),y)^2:
lprint( simplify( g(x,y) ) );

1+y^2+x^2

h := (x,y) -> sqrt( 1 + x^2 + y^2):
lprint( int(int( h(r*cos(t),r*sin(t))*r, r = 0..1), t = 0..2*Pi) );

4/3*Pi*2^(1/2)-2/3*Pi

by: gm